Talk:Riemann zeta function/Archive 1

Archive 1

Pronunciation

Is there a wikipedia policy for where pronunciations should appear? I dislike the redundancy of placing the pronunciation of Riemann both at this article and also at Bernhard Riemann. If a user wants to know the pronunciation they can simply follow the name link to find it. - Gauge 04:44, 2 September 2005 (UTC)

Also, pronunciations should probably be in International Phonetic Alphabet form, as English speakers could pronounce words very differently depending on the dialect. - Gauge 04:48, 2 September 2005 (UTC)

Remove the pronunciation from this article. linas 14:56, 2 September 2005 (UTC)

Riemann zeta function

I'd like to hold a survey regarding the article Riemann zeta function, to help determine its general comprehensibility and identify areas where it may be incomplete. Please indicate your perceptions of the article below, and feel free to expand the survey or article as you see fit. ‣ᓛᖁᑐ 21:07, 9 September 2005 (UTC)

The only point I could see this survey having is to determine which users are versed in some complex analysis and those who are not. See also my rant here. Dysprosia 07:57, 11 September 2005 (UTC)

Comprehensibility

Do you currently understand this article?

Yes

No

1. ‣ᓛᖁᑐ 21:07, 9 September 2005 (UTC)
2. I got a bit lost about halfway through the second paragraph. I also have no idea why the Euler Product Formula has to be proven in the middle of the article. And the bottom of the article is very difficult to follow with only college sophomore math skills. Avocado 00:56, September 10, 2005 (UTC)
3. — Xiong熊talk* 02:20, 2005 September 12 (UTC)

Comment

1. I agree with the fact that the proof of the Euler's product formula should not be there. Oleg Alexandrov 04:10, 11 September 2005 (UTC)
2. I understood most of it up to the point it started talking about Mellin transforms (which is the point at which I lost the will to follow all the links I didn't understand). Before that point, I think the Euler's product proofs should be moved out (but linked to), and a bit more should be said (or linked to) about the "importance of the zeros", in particular what or how path integrals relate to the prime counting function. Hv 11:03, 11 September 2005 (UTC)

---

If not, do you feel you could understand it after following its internal links?

Yes

1. ‣ᓛᖁᑐ 21:07, 9 September 2005 (UTC)

No

1. Er, well, I found Analytic continuation and Meromorphic function absolutely incomprehensible. I think that if I had time and energy to wade through a huge tree of links it might begin to make a bit more sense, but the motivation is a bit lacking ;-) . On the other hand, I doubt that reading through all the other articles Wikipedia has on math would make it possible to actually follow the mathematical equations and transformations. Avocado 00:56, September 10, 2005 (UTC)

• Er, well, this article could use improvement. But what's go me stumped is, if you don't know what a meromorphic function is, why in the world are you interested in the Riemann zeta? To answer my own question: probably because of the Clay Institute prize. In which case, the answer is that we need a section, or maybe even a whole article, describing "why the Riemann zeta is important to mathematics", and said article would not make use of formulas at all.

I say this because I object to the idea of somehow "simplifying" this article; its already too simple in many ways, because it is already quite lengthly, while failing to even touch on many important properties and relations. Personally, I would like to see some of the proofs and derivations moved to something in the style of Category:Article proofs. -- linas 17:03, 11 September 2005 (UTC)

Completeness

The article's lead section states the Riemann zeta function is "of paramount importance in number theory". From reading the article, do you understand why this function is important?

Yes

No

1. ‣ᓛᖁᑐ 21:07, 9 September 2005 (UTC)
2. Sort of. It has something to do with prime numbers (which I know are a knotty number theory problem) but I'm pretty sure I couldn't figure out from this article what exactly the connection is. Avocado 00:58, September 10, 2005 (UTC)

Removed text

I removed the following text:

The Möbius function also relates to the zeta function and Bernoulli numbers in the coefficients in series expansion of ${\zeta (n+2)} \over {\zeta (n)}$ with the formula

$\sum _{d|n}\mu (d)d^{2}$

for which A046970 gives values for the first 60 n.

I couldn't understand what it was trying to say. Its trying to describe some dirichlet series maybe??. linas 00:46, 21 December 2005 (UTC)

I spose i was trying to justify the oeis having a ref to this article. Numerao 21:33, 29 December 2005 (UTC)

An easier proof (for the layperson)

I found this section very helpful for understanding the connection between the Zeta function and prime numbers. BringCocaColaBack 11:29, 13 January 2006 (UTC)

Question moved from article

Following question from an anon contributor moved from the Globallly convergent series section on the article page. Gandalf61 08:35, 20 April 2006 (UTC)

Why is there a function of s only (zeta of s), that equals a sum which leaves behind a function of s and x?

That question applies more properly to the formula in the section above Globally convergent series, called Series expansions, which contains the formula:

$\zeta (s)={\frac {1}{s-1}}-\sum _{n=1}^{\infty }(\zeta (s+n)-1){\frac {x^{\overline {n}}}{(n+1)!}}$

What's that 'x' doing there? -GTBacchus^(talk) 13:38, 20 April 2006 (UTC)

I think that's supposed to be an s not an x, then it would look almost right (?). You can check correctness by going to the article on the Hurwitz zeta function, then look at the section called "Taylor series", where it mumbles about derivatives, and with a few minor substitutions (e.g. y=-1), you should be able to derive the above (with an s where x now stands). linas 04:16, 21 April 2006 (UTC)

Series Expansion

In the series expansion section, it's written that "Another series development valid for the entire complex plane is.." I can't figure out what the variable 'x' is supposed to be in the expansion that follows. x=s? —The preceding unsigned comment was added by 12.208.117.177 (talk • contribs) .

Ok, that's twice it's come up; I've changed it from x to s. -GTBacchus^(talk) 05:54, 11 May 2006 (UTC)

Hyphens

Is this function called the zeta-function or the zeta function? The article uses both. toad (t) 12:10, 10 February 2006 (UTC)

Some anon editor ran around adding hyphens recently, which no one reverted. Not just here, but in half-a-dozen articles. :-( 15:05, 10 February 2006 (UTC)

Zeta-function refers to all zeta function is general. But in this case its just Riemann zeta function. -- He Who Is^{[ Talk ]} 01:59, 29 June 2006 (UTC)

Surprising omission

No graphic of the graph in the complex plane? Surely the article should include one or perhaps three; real part, imaginary part and absolute value, as is the standard on Mathworld. Soo 14:17, 16 July 2006 (UTC)

lucky this is WP and not planetmath, or you would've gotten hooted at big time. anyway, yeah, it would be nice to have some kind of graphic. Numerao 20:18, 17 July 2006 (UTC)

Third moment of the Riemann zeta function?

The article titled 42 (number) says:

It is believed to be the third moment of the Riemann zeta function, based partially upon evidence from quantum mechanics.

I don't know what this means. Here's a guess:

$\int _{(-\infty ,\infty )}s^{3}\zeta (s)\,ds=42.$

I'm accustomed to the definition of momnets of probability measures; if ζ were a probability density function then the integral above would be the third moment of the corresponding probability distribution. But ζ is negative in some places, and from the way ζ(s) blows up at s = 1 it seems we'd have to be thinking of a Cauchy principal value or something like that.

Can someone make the article's statement clearer? Michael Hardy 17:52, 5 July 2006 (UTC)

• • • • • I have edited the article on "42" to make clearer the connection to moments of the zeta-function. It is actually the "sixth moment" that is interesting, and the moments are usually defined

$I_{k}(T)={1 \over T}\int _{0}^{T}\left|\zeta ({1 \over 2}+it)\right|^{2k}\,dt.$

When k=3 you get the "sixth moment". The constant 42 comes up as a scaling factor in a conjecture by Conrey & Ghosh for the leading order term of this integral. .—Preceding unsigned comment added by 74.74.128.91 (talk • contribs) *****

I've exchanged some email with John Baez, the mathematical physicist who has edited Wikipedia articles as user:John Baez, and he reports that he cannot access Wikipedia because he is in China. He wrote:

“ So, feel free to post this comment for me:

I don't know anything about the "3rd moment of the Riemann zeta function", but perhaps what's meant is the 3rd moment of the distribution of spacings between zeroes of the Riemann zeta function. There's a lot of evidence relating the distribution of these spacings to the distribution of spaces between eigenvalues of a large random self-adjoint matrix. For lots more, try this:

http://www.maths.ex.ac.uk/~mwatkins/zeta/bump-gue.htm

and for general connections between the Riemann zeta function and quantum mechanics, try:

http://www.maths.ex.ac.uk/~mwatkins/zeta/physics1.htm

Best, jb

”

My wild guess seemed so implausible that I'm both relieved to hear that it's wrong and pleased to hear that this otherwise implausible-seeming statement can be construed in such a way that it makes sense. Michael Hardy 16:20, 20 July 2006 (UTC)

graph of zeta

I stored in commons a graph of zeta (x) with -20 < x < 10.

see

Riemann zeta function for real -20 < s < 10. The green graph is 100*zeta(x), -13 < x < -1

Perhaps it can go into the article

--Brf 10:05, 31 August 2006 (UTC)

Critical Strip

Can someone add an explanation about the critical strip? This term's definition is nowhere to be found in Wikipedia. Thanks! By the way, the whole article is fine and readable; everybody with a college degree will understand at least the basics. Hugo Dufort 08:28, 14 November 2006 (UTC)

Okay, done that. Gandalf61 09:54, 14 November 2006 (UTC)

Thanks a lot! The added information greatly helps understanding some key concepts. Every specialized term needs to be defined, unless people not familiar with the subject will be lost on the way. Even before the arid math proofs! Hugo Dufort 04:16, 15 November 2006 (UTC)

Euler product

I added an expanded form of the zeta function I got from Marcus de Sautoy's "Music of the Primes", because I think it helps to visualise exactly what the product is about. Comments? DavidHouse 21:21, 26 December 2006 (UTC)

An alternative elementary formulation for Zeta function

In a technical report entitled "[http://cswww.essex.ac.uk/technical-reports/2005/csm-442.pdf An elementary formulation of Riemann’s Zeta function]", myself (Riccardo Poli) and Bill Langdon provided a very simple proof that, for $\Re (s)>1$, Riemann's Zeta function can be written as $\zeta (s)=1+\sum _{k=1}^{\infty }a_{k}(s)p_{k}^{-s}$ where $a_{k}(s)=\prod _{j=k}^{\infty }(1-p_{j}^{-s})^{-1}$.

We are not experts in number theory, but we have searched widely and also asked several mathematicians: it appears that our rewrite is new. These people tell us that this is useful formulation. So, we were wondering whether it would make sense to include it in the article.

Yes, this follows from the fact that

$a_{k}(s)=\sum _{m\in \mathbb {P} _{k}}m^{-s}$

where $\mathbb {P} _{k}$ is the set of integers whose smallest prime factor is greater than $p_{k}$. If this result is published in a textbook or refereed journal then it can be included it in the article. If not, however, then it falls under Wikipedia:No original research and cannot be included. Gandalf61 09:27, 27 October 2006 (UTC)

The paper mentioned above has now been published in arXiv.org in the Mathematics History and Overview section (math.HO/0701160). Perhaps the result could now be included in the article?

As Gandalf61 said, if the result is published in a textbook or refereed journal then it can be included it in the article. Papers in arXiv.org do not undergo peer-review, so they are usually not considered reliable sources. Hence, the answer is no. -- Jitse Niesen (talk) 15:04, 9 January 2007 (UTC)

I for one welcome our Rieman zeta function wielding mathematical overlords.

I want to add a personal support for the writers of the article. Even though the article necessarily is largely technical, the lead-in paragraph adequately establishes the backround of the function for laymen. -- Cimon Avaro; on a pogostick. 08:51, 12 February 2007 (UTC)

1/(s-1) should be s/(s-1) in the rising factorial series

209.226.117.54 16:08, 17 February 2007 (UTC) Jacques Gélinas

Gentler definition

Browsing through the talk page, it seemed to me that there had been quite a few complaints about the definition of from people not comfortable with analytic functions, or perhaps, with mathematics in general. I can definitely confirm that the so-called "introduction" to this article is too terse to be of any use. Other articles, such as Riemann hypothesis are much better in this regard. So this is certainly something that needs to be dealt with. For now, I have expanded the definition a bit, it remains a rigorous mathematical definition, so it's unclear to me how much happier would non-mathematicians be with it. Hopefully, it is somewhat gentler to those who are unsure about all the symbols and unfamiliar terminology, although to experts on Riemann zeta function it may appear to be perhaps a little too easy. I do want to point out that Enrico Bombieri, in the description of the Riemann hypothesis in the Millenium Prize book starts by mentioning that the Dirichlet series for $\zeta (s)$ is defined only for large $s$, and then explains that it is analytically continued. I definitely feel that it's not something to be taken lightly, especially since analytic continuation of general Artin L-functions is still unknown, and of course, by no means obvious! Perhaps, it would make sense to expand the definition even more, it is a judgement call (or an editorial decision), so I would wait to hear the reaction.

Incidentally, I think that in line mathematical formulas do not look very good in this case, but since it's a highly emotional issue for at least some users, I tried to preserve them. Arcfrk 06:04, 10 March 2007 (UTC)

Better Non-technical Description

For those who get lost in this Wikipedia article, I have found the following link to be the clearest description in layman's terms. Those who authored and are maintaining this Wiki article may want to read this to understand how a complicated math concept can be described in normal conversational English:

http://seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php Overlook1977

This is definitely a worthy link! On the other hand, encyclopedias are not written in normal conversational English, and very much on purpose so. Arcfrk 17:25, 16 April 2007 (UTC)

I glanced at if for a second or two. Is there anything in this link that actually says what the zeta function is?? If not, I certainly wouldn't say it's clear to either lay persons or anyone else. Michael Hardy 20:45, 16 April 2007 (UTC)

The article (which is 3 pages long, if you did a quick glance you probably missed the meat of the article on page 2) discusses the relevance of the Riemann Zeta Function to prime numbers. Those not familiar with advanced mathmatics or number theory are probably more interested in the significance/importance of the function vs. how it is technically derived.

Im just trying to provide a non-technical explanation of the significance of the RZF, which I am still working on understanding. The article isn't 100% relevant but it does provide some insight. When I better understand the RZF myself I will contribute to the main article. The problem with this and other advanced math articles on Wikipedia is they fail to describe the overall big picture. For example, you could explain that the volume of a sphere is 4/3¶r3. But you could also say it simply the measurement of the space inside a sphere. You couldnt take the latter description and do a calculation with it, but not everyone is interested in the details. Some of us want the big picture. Overlook1977

Connection to prime numbers?

This article needs more information on how the zeta function is connected to prime numbers. My understanding of this function and the Riemann hypothesis is that this function and prime numbers are very deeply connected, but I can not find on Wikipedia or anywhere else an explanation as to why. Sloverlord 12:36, 17 May 2007 (UTC)

Have you tried Prime number theorem? Arcfrk 19:34, 17 May 2007 (UTC)

Gamma function reference

The Mellin Tranform section has $\Gamma (s)$. Could "where $\Gamma (s)$ is the Gamma function" be added?

Sure. That was noted before, but it won't hurt to repeat it, so I added it. However, you should feel free to make such edits yourself. The "edit this page" button at the top of the article is there for a reason! Cheers, Jitse Niesen (talk) 08:23, 20 May 2007 (UTC)

!!! download now !! ??

Is this vandalism? At the bottom of the page there's a link to a .gz file. Even so, it should be reworded. '''T'''''o''__m__ 17:34, 7 November 2007 (UTC)

Specific Values - wrong place?

I thought the Riemann zeta function referred to complex numbers - what is the justification for including the harmonic series, the Basel equality, and other zeta functions that don't involve complex numbers here? 24.61.112.3 15:12, 2 December 2007 (UTC)

The Specific Values section contains examples of series for natural numbers only. Don't these belong in an article about Zeta functions "in general"? The sudden transition from discussing natural number constants to the "Zeta zeros" - and hence complex numbers of the Riemann Zeta function - is bizarre and misleading to say the least. Michaelmross 14:53, 21 January 2007 (UTC)

I disagree. The article says right at the beginning that it's defined at all complex arguments except the number 1. Nothing in the section on specific values contradicts that. Also, it's NOT about "zeta functions in general"; it's about one function---the Riemann zeta function. Michael Hardy 03:20, 22 January 2007 (UTC)

... and now I see that that section does not give values only at natural numbers. Thus your comments above are what is misleading. Michael Hardy 03:32, 22 January 2007 (UTC)

I find it sad that a well-intended comment is misunderstood and then labeled as misleading. I didn't say the topic was about "zeta functions in general" - I clearly suggested that natural number series might belong in a *topic about zeta functions in general*. Because what I'm saying about this section of the article is that it goes from something general about natural numbers that a layperson like myself can understand to something very specific concerning complex numbers and the zeta zeros. And to me - a non-mathematician - this is very confusing. I would like to see a layperson's distinction between a generic zeta function and a Riemann zeta function. This will be my last comment on the matter, so there's no need to flame me any further. Michaelmross 20:12, 22 January 2007 (UTC)

Nobody flamed you. I find your remarks confusing. You contrast "something general about natural numbers" with "something very specific concerning complex numbers", but in fact it is the natural numbers that are specific and complex numbers that are general. As for "a layperson's distinction between a generic zeta function and a Riemann zeta function", I think you might find any definition of "zeta functions in general" to be somewhat more abstruse than an account of the Riemann zeta function. Anyway, you're being unclear; I have a hard time trying to figure out what you're saying. Michael Hardy 22:47, 22 January 2007 (UTC)

Incorrect Statements

The approximation by Gergő Nemes in "The functional equation" does not work when

1. s is close to 0
2. s has a large imaginary part (in fact, according to the formula zeta has only the trivial zeros)
3. s gets very negative

In other words, it mostly makes sense as an approximation of zeta for large negative values. For these values, the derivation is simply plugging in Stirling's formula into the functional equation, which only serves to complicate the expression. —Preceding unsigned comment added by 64.3.169.42 (talk) 21:15, 3 January 2008 (UTC)

Continuation of $\zeta$ function in the domain $Re(s)<=1$

I must ask a naive question to the author.I do not understand how the continuation of the $\zeta$ function is performed when variable s is such as $Re(s)<=1$.One was speaking of an integral formula,but the formula was never shown.The only integral formula that I saw was :$\zeta (s)={\frac {1}{\Pi (s-1)}}{\frac {1}{e^{-i\pi s}-e^{i\pi s}}}\int _{+\infty }^{+\infty }{\frac {(-x)^{s}}{x(e^{x}-1)}}dx={\frac {1}{\Pi (s-1)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}dx$ where $\Pi$ is the well-known Pi-function. But the last integral converges for $Re(s)>1$ only,so the first integral has a meaning only on this condition.So I do not understand the fact of how the $\zeta$ function initially defined by a serie,is continued for $Re(s)<=1$.I have tried to go to the source (Riemann's work in english version),but I still not understood how the continuation was performed.So I ask to the author of the article to bring the light on this (important) point of the subject.

Excuse me not to sign my message,but I am not involved among the users. date 08.01.2008 —Preceding unsigned comment added by 90.15.195.191 (talk) 16:51, 8 January 2008 (UTC)

Removing "Garbage"

I don't know how I missed them before, but the sections purporting to evaluate zeta(2) and zeta(4) can, perhaps, be best characterized as unencyclopaedic garbage, the word that they seem to be using quite a bit. "Zeta constant" already offers a thorough coverage of special values of zeta. The "proofs" themselves are about as tortured as one can imagine, and blatantly fail WP:NOT. I got rid of them. Arcfrk (talk) 03:45, 3 March 2008 (UTC)

Discrete zeta function graph

What is the point of this graph in the middle of the article, when the article does not mention it even once? T.Stokke (talk) 21:05, 1 March 2008 (UTC)

Yes, it's not really clear what that graph represents. Certainly not the partial sums of the series defining ζ(2). I've commented it out for the time being, it might be relevant but it's not immediately obvious from the caption. Arcfrk (talk) 03:52, 3 March 2008 (UTC)

Trivial zeros - again

Years ago I did a course in complex analysis. I got a bare pass, which doesn't seem sufficient to understand the section on trivial zeros. Could someone explain explicitly why the negative evens are zeros?RayJohnstone (talk) 15:50, 28 March 2008 (UTC) —Preceding unsigned comment added by RayJohnstone (talk • contribs) 15:47, 28 March 2008 (UTC)

I think the easiest way to see it is using the reflection formula: $\zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)$

If s is a negative even integer, then $\sin \left({\frac {\pi s}{2}}\right)=0$, so the whole thing is 0. -GTBacchus^(talk) 15:55, 28 March 2008 (UTC)

zete(0)

As I understand it: zeta(0) = 1/1^0 + 1/2^0 + 1/3^0... = 1 + 1 + 1... = infinity != -1/2

Where do I wrong? 84.108.198.129 (talk) 23:35, 2 September 2008 (UTC)

Now I see the answer above. Sorry... 84.108.198.129 (talk) 23:43, 2 September 2008 (UTC)

How does ζ(0)=-1/2?

Shouldn't ζ(0)=infinity?
$\zeta (0)={\sum _{n=1}^{\infty }}{\frac {1}{n^{0}}}={\sum _{n\geq 1}^{\infty }}{\frac {1}{1}}=1+1+1...=\infty$

05:47, 19 April 2008 (UTC)

The series definition of the zeta function doesn't apply unless the real part of z is greater than 1. -GTBacchus^(talk) 10:06, 19 April 2008 (UTC)

Yes this is also addressed in the article.T.Stokke (talk) 14:01, 26 April 2008 (UTC)

Still...

How does ζ(0)=-1/2? —PrecedPMajer (talk) 16:21, 8 September 2008 (UTC)ing unsigned comment added by Anon126 (talk • contribs) 05:59, 15 May 2008 (UTC)

As far as I know, the easiest way of computing ζ(0) uses the representation at the end of section 4.1 :

$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }\left({\frac {x^{s-1}}{\exp(x)-1}}\right)\ dx.\!$

Starting from this, one computes ζ(s) for s=0, and more generally for s=-n with nonnegative integers n, in terms of the residue at x=0 of

${\frac {x^{-n-1}}{\exp(x)-1}}$,

which relates ζ(-n) with the Bernoulli numbers:

$\zeta (-n)=-{\frac {B_{n+1}}{n+1}}$.

Note also that the functional equation relates ζ(n) and ζ(1-n), whence you can compute the sum of the series of ζ(n), for even n>1. Conversely, if you already computed in another way the values of the sum of the series, you can deduce, of course, the values of ζ(n) for n<0 (but not for n=0, as you observed). See also the article on the Hurwitz zeta function for a generalization. PMajer (talk) 09:24, 22 May 2008 (UTC)

In my mind, the easiest way to make sense of this is to recall the relationship between the zeta function and the Dirichlet eta function:

$\zeta (s)={\eta (s) \over {1-2^{1-s}}}$:

If you plug 0 into the alternating sum for the eta function, you have 1 - 1 + 1 - 1 ... which would lend itself to being equal to 1/2, which, when divided by (1 - 2), yields -1/2. -Xylune (talk) 11:32, 12 August 2008 (UTC)

Of course, this does not make any sense... Well, sorry, I mean, one should justify the equality with the Cesaro summation. (remember that, playing breezly with non convergent series, one can easily produce many identities like "1=0" etc..) PMajer (talk) 16:21, 8 September 2008 (UTC)

Too math or not too math

This article shows a kind of a common problem of mathematical articles. It seems that a lot of people are interested in the Riemann zeta function, without beeing so interested in understanding the elementary underlying facts, such as, what is a complex number, or what is the sum of a series, etc. This makes sense, of course, for the sake of a general information (in the same way, I am interested in music even though I cannot play). But in this case one should not complain if an article looks too technical, or if things are not made easier than possible, and one should not be surprised if he or she can't go beyond a row picture of the facts, because it is natural. The point is simply that a complete explanation with all details, e.g. about the zeta function, would require a good part of an elementary course of complex variables, what is beyond the scope of thes articles. On the other hand, there is a minority, made of users with a mathematical culture, that go to a math article looking for a more technical, precise and usually very short information (tipically, in areas of mathematics different from their own). For them wikipedia has become quite a useful tool, so I think it should be a mistake to keep an article to the level of the mean users. Maybe a good thing should be to keep disjoint the general and the specialistic part...PMajer (talk) 17:48, 8 September 2008 (UTC)

zeta(-2)

zeta(-2) = 1/1^-2 + 1/2^-2 + 1/3^-2 + ... = 1 + 2^2 + 3^2 + ... = infinity !=0

Someone is wrong - me, or the rest of the world. Who? (and why?) 84.108.198.129 (talk) 23:43, 2 September 2008 (UTC)

The first sentence in the definition explains why you're wrong. The real part of -2 is not greater than 1. Ben (talk) 04:20, 3 September 2008 (UTC)

This is because of analytic continuation: a formula valid for some values may not make sense for others. As a simpler example, consider 1 + x + x² + ... = 1/(1-x) which makes sense for |x| < 1. The series definition does not make sense for |x| > 1, but the formula 1/(1-x) does. Richard Pinch (talk) 23:20, 8 September 2008 (UTC)

dubious fractal dimension

How can a subset of a straight line have a fractal dimension greater than 1? Or does it mean that the Riemann hypothesis is false?--Yecril (talk) 23:08, 6 October 2008 (UTC)

Surprisingly enough, it seems to be correct. See this summary. Richard Pinch (talk) 17:12, 7 October 2008 (UTC)

...

Does anyone agree that the "Applications" section is a little silly? The connection to physics is sparse, and no connection to the use of the Riemann-Zeta function is actually motivated by it (would you actually need to use it to show that sum diverges? Surely you want to show something else). Njcnjc (talk) 03:03, 21 October 2008 (UTC)

Can anyone elaborate on the "prime numbers" section? It says "this is a consequence..." without giving even a hint or flavour of why. --Doradus 21:25, Sep 17, 2004 (UTC)

I've added what I believe to be a standard treatment of why the two expansions are equivalent. Do any real mathematicians want to do a better job? -- The Anome 23:27, 17 Sep 2004 (UTC)

I've turned the "vigorous handwaving" into an actual proof and have made a number of further changes and additions. Gene Ward Smith 08:42, 29 Jan 2005 (UTC)

I feel readability is fine

There will always be a long standing question on the readability of this article; I feel the need to put my two cents in. I came to the page after reading probably 20-30 pages on abstract math, no formal training in mathematics beyond Differential Equations and Vector Calculus in college. I found it to be sufficiently readable that I was able to "get the gist" of what the Riemann Zeta function is and why it is important. No, I could not do a single calculation regarding it, but from what I gather, the masters of the topic have trouble doing more than the most basic proofs regarding the topic - so is it such a surprise that I have trouble?

Fundamentally, Riemann Zeta is not simple. Not by a long shot. But that doesn't mean its unimportant; it doesn't mean that mathematicians who can leverage the information should be denied such a tool. There's still truly trivial content on wikipedia; lets leave the profound content (like Riemann Zeta) in place. 199.46.245.230 (talk) —Preceding undated comment was added at 17:04, 26 November 2008 (UTC).

re: Application detail

I don't feel there is adequate discourse of using the reciprocal form. Specifically, readers will wonder why it is used when the sum of positive integers from 1 thru N is well established as N*(N+1)/2. Is the manipulation to be able to apply specialized analysis forms or ??--Billymac00 (talk) 03:17, 27 January 2009 (UTC)

Question on Trivial Zeros

The trivial zeros do not seem to yield zeros:

For example: $\zeta (-2)=1+{\frac {1}{2^{-2}}}+{\frac {1}{3^{-2}}}+\cdots =1+2^{2}+3^{2}+\cdots$;

which yields infinity.

I will appreciate if someone lets me know where I am mistaken.

As the article says, the power series only defines the Riemann zeta function for arguments x > 1. You must use the function's analytic continuation to evaluate it elsewhere. Fredrik Johansson 18:46, 26 August 2006 (UTC)

The "trivial zeros" are trivial because of the functional equation. There has to be zeros of the zeta-function to "cancel" the poles of the gamma-function, otherwise the zeta-function would not be analytic in the left half-plane.—Preceding unsigned comment added by 74.74.128.91 (talk • contribs)

Though with enough careful parsing of the article the fact that numbers with Re(s) < 1 are not in the domain of the power series is buried in the article, it is in no way clear. It took me forever to understand that that is what ya'll meant. This issue about the trivial zeros is subtle point that no one really explains well on any of the online sources. Why not start the article with the power series definition and a quantifier that explicitly says that values with Re(s) < 1 are not valid for this function, then put the analytical function next to it and use a quantifier to show that its domain covers the entire complex plane? Or at the very least add one sentence clarifying this point in no uncertain terms. I'm about 1/2 a step from adding it myself but I don't feel very qualified to talk about this topic....but if someone else doesn't do it, or they can't think of a good reason why it shouldn't be done...I'm gonna have to.

128.97.68.15 (talk) 17:25, 1 February 2008 (UTC)

By "careful parsing", do you mean reading the first and second sentences of the section labeled "Definition"? It states that the Dirichlet series converges for s > 1 and is only defined for other s through analytic continuation. "If everything else fails, read the definitions". Arcfrk (talk) 20:07, 1 February 2008 (UTC)

Although it's in the definition, someone who's not well-versed with terms like "analytic continuation" will probably skip the qualifiers, and remain serenely uninformed until they get to the assertion that s = -2, -4, -6, and so on are trivial zeros. It would be useful to repeat that particular qualifier there (s > 1), I think, and perhaps point back to that definition. I'll come back in a couple of days and do that unless someone protests. BrianTung (talk) 00:38, 15 March 2009 (UTC)

Riemann zeta function

Based on a false premise, don't you think? Fergananim (talk) 20:51, 5 February 2009 (UTC)

Huh? -GTBacchus^(talk) 03:36, 15 March 2009 (UTC)

Moved from article

The following snippet appears to be meaningless. What is s? What is the significance of the variable x? A broader question is: is this formula significant enough to be in the article? If so, could somebody please correct it and give it enough context that it becomes meaningful and manifestly notable for inclusion in the article. Sławomir Biały (talk) 03:05, 9 July 2009 (UTC)

Expansion of the logarithm of ζ(s) on the critical strip

${\begin{aligned}\pi {\frac {dN}{dx}}(x)&={\frac {1}{2i}}{\frac {d}{dx}}\left(\log \left(\zeta \left({\frac {1}{2}}+ix\right)\right)-\log \left(\zeta \left({\frac {1}{2}}-ix\right)\right)\right)\\[8pt]&{}\qquad -{\frac {2}{1+4x^{2}}}-\sum _{n=0}^{\infty }{\frac {2n+{\frac {1}{2}}}{\left(2n+{\frac {1}{2}}\right)^{2}+x^{2}}}\end{aligned}}$

here dN(x)/dx is just the derivative (as a distribution) of the number of zeros on the critical strip 0 < Re(s) < 1. A proof of this can be found on a work by Guo (see references).

- the derivatives are all respect to 'x'

- The formula is an expansion of the logarithm of $\zeta (1/2+is)$ on the critical strip, as you can see it reproduces all the poles of $\zeta (1/2+is)$

- ${dN}{dx}$ is just the expression $\sum _{\gamma }\delta (x-\gamma )$ , whith 'gamma' being a sum over the imaginary part of the zeros --Karl-H (talk) 09:45, 9 July 2009 (UTC)

I see you've gone ahead and restored the context, with fixes. Thanks for that, but I still see the second part of my question as entirely unaddressed, namely does this formula actually belong in the article at all? It was sort of vaguely referenced before, and now it is entirely unreferenced. But I get the feeling that this isn't a typical formula that one can dig out of any textbook (or mathematical encyclopedia, for that matter). And so the question remains: is inclusion of this formula consistent with the spirit of the requirements of WP:UNDUE weight? If so, then it should be possible to build a more meaningful context around the formula in question: What is it good for? Why should the reader care about it? Otherwise, the disputed content should I think be removed, as Wikipedia is WP:NOT an indiscriminate collection of information. Sławomir Biały (talk) 18:40, 10 July 2009 (UTC)

this formula is just an expansion of the logarithmic derivative of Zeta function on critical strip, it is interesting (just my opinion) since it involves a sum over the poles of Riemann zeta, i think i found in the paper refereed before or in another context in papers talking about Gutzwiller Trace formula and Riemann Hypothesis, perhaps this formula would suit better into an article about Riemann Hypothesis or Gutzwiller trace, as you wishes --Karl-H (talk) 20:19, 10 July 2009 (UTC)

Zeta function converges for Re(s) > 1 and diverges elsewhere??

If so, it wouldn't have had any zero on critical strip Re(s)=1/2, neither any trivial zero.

Clearly, '>' should be replaced by '<'.

See paragraph 1. —Preceding unsigned comment added by 151.53.136.190 (talk) 21:02, 12 August 2009 (UTC)

The zeta function itself is defined for all complex numbers except 1. The series $\sum _{n=1}^{\infty }n^{-s}$ is only convergent for Re(s) > 1, the zeta function is defined elsewhere in another way (analytic continuation). — Emil J. 09:54, 13 August 2009 (UTC)

Negative even integers don't work

The article says that all negative even integers are trivial zeroes, but the value of the zeta function is infinite for them, not 0.

1+4+9+16+..., 1+16+81+256+..., etc. all diverge to infinity. —Preceding unsigned comment added by 75.28.53.84 (talk) 14:51, 15 August 2009 (UTC)

Did you read the definition? The series is only for Re(s) > 1. --Zundark (talk) 15:35, 15 August 2009 (UTC)

It said that all negative even integers are trivial zeroes. --75.28.53.84 (talk) 15:43, 15 August 2009 (UTC)

Yes, and that's correct. See my reply to you at User talk:Zundark#Riemann zeta function. --Zundark (talk) 16:14, 15 August 2009 (UTC)

The "definition" section does NOT make it clear that the infinite-series expansion only applies to Re(s) > 1. The definition says, "The Riemann zeta function ζ(s) is the function of a complex variable s, initially defined by the following infinite series...". It puts no restriction on "s." The next sentence, "Leonhard Euler considered this series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1," also does not make this restriction clear. It just talks about two mathematicians who "considered" and "extended" the formula. All that sentence suggests to the reader is that before Chebyshev, the formula didn't even apply to Re(s) > 1. The definition needs to be clarified. -LesPaul75_talk 05:43, 28 September 2009 (UTC)

Changed definition

Given the above endless confusion about the definition of the series, the definition of the function defined by the series and the definition of the zeta function itself, I've changed the wording of the definition to something that I hope is slightly clearer. I attempted to make clear what people on the talk page were saying: that the zeta function is the analytic continuation of the function defined by the series given. Please take note that I have NO UNDERSTANDING AT ALL of analytic continuations. I merely changed the definition to emphasize what has been repeatedly said on this talk page. If I got it wrong, please feel free to correct it. mkehrt (talk) 09:03, 7 October 2009 (UTC)

It's better, but in my opinion it's a little redundant now. There is "Re(s) > 1" stamped all over the definition in five different places. There's no need to beat the reader to death. Why not just make it match the definition here: Riemann_hypothesis? That definition gets right to the point in just one sentence. -LesPaul75_talk 10:05, 15 October 2009 (UTC)

History?

The Euler product formula is from about a hundred years before Riemann was born. Is it known who was the first to consider this function? It seems Chebyshev's work connecting prime numbers to this function was before Riemann. Why is it called the Riemann zeta function? Was Riemann the first to define it for complex numbers? A section on history would be nice. Regards, Shreevatsa (talk) 14:57, 18 March 2009 (UTC)

The article states that the function was named after Riemann because "he introduced it"... Is that the case, or is it as Shreevatsa says, that he was the first to define it for complex numbers? —Preceding unsigned comment added by 24.12.13.8 (talk) 19:07, 15 October 2009 (UTC)

See On the Number of Primes Less Than a Given Magnitude, which is already linked to in the text. Fredrik Johansson 16:51, 18 March 2009 (UTC)

Flaw in the argument regarding the infinitude of primes

The argument: (harmonic diverges) -> (euler's formula predicts infinitely many primes) is flawed since Euler's formula depends on there being infinitely many primes to sieve the infinite sum of the riemman-zeta function to '1' —Preceding unsigned comment added by 99.73.17.9 (talk) 06:17, 7 April 2010 (UTC)

No, the derivation of the formula only uses the unique factorization property, it makes no difference whether there are finitely or infinitely many primes (in fact, the derivation is easier if there are only finitely many primes, since some issues of convergence disappear).—Emil J. 10:16, 7 April 2010 (UTC)

I'm not sure I agree with this. If there are finite primes, then I don't think the sieve can reduce zeta(2) (with its infinite terms) to a finite amount of terms. but i could be wrong, since all infinities involved are countable. —Preceding unsigned comment added by 99.73.17.9 (talk) 02:47, 8 April 2010 (UTC)

First, why do you keep talking about a sieve? There is no sieving involved. Second, did you actually read the derivation of Euler's formula, or are you just making your impression up? If there are finitely many primes (lets call them p₁,...,p_k), the Euler product is still a finite product of infinite sums, which evaluates to an infinite sum by distributing the product over the sums:

$\prod _{p}{\frac {1}{1-p^{-1}}}=\prod _{p}\sum _{m=0}^{\infty }p^{-m}=\prod _{i=1}^{k}\sum _{m=0}^{\infty }p_{i}^{-m}=\sum _{m_{1},\dots ,m_{k}}(p_{1}^{m_{1}}\cdot \dots \cdot p_{k}^{m_{k}})^{-1}=\sum _{n=1}^{\infty }n^{-1}.$

As I already wrote, the only thing used here is the unique factorization property of integers, which guarantees that the products $p_{1}^{m_{1}}\cdot \dots \cdot p_{k}^{m_{k}}$ exhaust all positive integers, each one once.—Emil J. 10:24, 8 April 2010 (UTC)

I think you haven't read the method Euler used. He certainly does 'sieve' out the primes. First he sieves out multiples of 2 with 1 - 1/2^s, then he sieves out multiples of 3 with 1 - 1/3^s, etc. He does this for all primes. This is just like an Erosieve. zeta(1) diverges. You start with an infinite sum. You sieve out all the primes. You get a product for all primes. To extend the sieving method all the way ad infinum and be left with a finite result assumes that there are an infinite amount of primes to sieve out. So the fact that we have a divergence equal to a product of all primes rests on the assumption that there were an infinite amount of primes to begin with. I see now why you didn't understand me to begin with since you were unfamiliar with the content of this page and unaware that a sieving process was used to generate the formula. I now see I was right from the start. —Preceding unsigned comment added by 76.182.194.195 (talk) 02:39, 16 April 2010 (UTC)

If there are only finitely many primes, your "sieving" process stops after finitely many steps. There's nothing wrong with that. (Since you seem to like the sieving analogy: the Eratosthenes sieve also stops after finitely many steps if there are only finitely many primes, once you take care of the last prime there are no more numbers left in the sieve.) You seem to be having some kind of a mental problem with the trivial observation that an infinite sum can be decomposed to a finite sum (or product) of infinite sums. Furthermore, calling this derivation "sieving" is a bit of stretch. Extracting the term 1 − 1/2 is not achieved by deleting multiples of 2 from the infinite sum, but by splitting each even term 1/(2n) into a product of 1/2 and 1/n, distributing this 1/2 before the sum, and transferring this part to the other side of the equation, which means that the term in the sum corresponding to a given number is merged with the term corresponding to its odd part, it's not deleted. All of this is completely irrelevant anyway: I gave you a valid proof above, it makes no difference whether there is another proof which may or may not work; if you can't understand that, then I'm afraid that mathematics is not a field for you.—Emil J. 10:03, 16 April 2010 (UTC)

Oh, and by the way: the original Euler's argument which you call sieving is actually invalid, as it relies on repeated subtraction of divergent series (you can easily prove 0 = 1 using the same method). That's why derivations of the formula in modern literature follow the approach I outlined above (or something similar) instead, which guarantees that all terms in the sums and products are nonnegative, and therefore the manipulations are correct even if the results are infinite. (Alternatively, one may derive the formula for s > 1, and consider the limit as s → 1.)—Emil J. 13:36, 17 April 2010 (UTC)

Analytic continuation (again)

Only the original formula of Riemann zeta function is in the article. I believe it's very important to also include its analytic continuation formula too. It is a crucial piece of information. Also it would be nice to add in the article how this analytic continuation was found. I'm searching for this information myself so if somebody could do it it would be very appreciated. SmashManiac 20:15, 19 April 2007 (UTC)

By how it was found, do you mean history, or the actual derivation? The history is very short: Riemann's revolutionary paper (link in the text) defined analytic continuation. Riemann gave two proofs of the analytic continuation, one of them using an integral representation. Is that what you mean by formula? There should definitely be a place in Wikipedia where this is explained. There are also formulas for special values (negative integer s), and there is functional equation in case you need ζ(s) for Re s < 0 in terms of ζ(s) in the convergence region. Arcfrk 00:01, 20 April 2007 (UTC)

Hi Arcfrk,

Thanks for the history of derivation. Can you also provide the actual analyiticall continued formula for complex plane.

-Subhash —Preceding unsigned comment added by 12.144.36.2 (talk) 19:09, 24 May 2010 (UTC)

Formula in complex plane

Can some insert the actual RZF for complex plane. I found some formula at [1]. But I do not think I am competetive enought to validate the formula. -Subhahsh —Preceding unsigned comment added by 12.144.36.2 (talk) 19:59, 24 May 2010 (UTC)

There are many such formulas. Of those already used in the article, at least the formulas in sections "Laurent series", "Rising factorial", "Hadamard product" and "Globally convergent series" are valid for the whole complex plane.—Emil J. 16:14, 26 May 2010 (UTC)

Analytic continuation

For the given functional equation

$\zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)$

it follows that

$\zeta (2)=0$

but

$\zeta (2)={\frac {\pi ^{2}}{6}}$

Why are these values different? 67.185.99.246 02:40, 11 February 2007 (UTC)

The gamma function has simple poles at s = −n (n = 0, 1, 2, 3, ...), and so we have to be careful while evaluating

$\lim _{s\rightarrow 2}\;\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)$.

Using Euler's reflection formula for the Gamma function I get

$\zeta (s)=(2\pi )^{s}{\frac {\sin(\pi s/2)}{\sin(\pi s)}}{\frac {\zeta (1-s)}{\Gamma (s)}}$,

and from l'Hôpital's rule I get

$\lim _{s\rightarrow 2}\;\zeta (s)=-2\pi ^{2}\zeta (-1)$

From this and from

$\zeta (2)={\frac {\pi ^{2}}{6}}$

I get

$\zeta (-1)=1+2+3+\dots =-{\frac {1}{12}}$,

a result I think already known to Euler. —Tobias Bergemann 08:54, 11 February 2007 (UTC)

That works of course, thank you for that explanation. 67.185.99.246 08:21, 12 February 2007 (UTC)

Speaking of that formula, it actually doesn't help find riemann zeta function since it uses circular definition to define it (if you don't know riemann zeta of n or 1-n you can't use it). It can be used to define factorial if you isolate (-n)! and let n=-x. Note: letting x=0 or -1 will result in zero times infinity in the formula, so here is the heads up on those: 0!=1 -1!=+-infinity srn347 —Preceding unsigned comment added by 68.7.25.121 (talk) 06:33, 19 November 2008 (UTC)

I was likewise confused about the meaning of the functional equation. It says it's valid for all s (except 0 and 1)... but since the gamma function has poles for negative integers, the functional equation is actually meaningless for s = 2, 3, ..., _unless_ it is understood to involve limits in that case. Since the average reader might not be aware of this shorthand, I mentioned it in the article. If you can find a better way to say what I said, go for it. Kier07 (talk) 03:20, 24 August 2010 (UTC)

One can tell from your edit that you were confused. The functional equation makes perfect sense as an identity between two meromorphic functions, valid on the whole complex plane. I am not convinced that this level of detail is appropriate (Wikipedia is not a textbook), but for the time being, I've emended the explanation. Arcfrk (talk) 04:28, 24 August 2010 (UTC)

Joe test failure

Disclosure: I am the son of a world-famous mathematician -- indeed, in her later life a number theorist. However, she mated with a whiskey-soaked advertising copywriter and I was raised on a farm by psychopathic semi-deaf-mutes shunned by their neighbors for the worldly sin of using electricity and automobiles. Mother did not rescue me from this Appalachian pastoral idyll for many years, whereupon she made my bedtime tales out of graph theory.

So, I have only half the genes and half the nurture. I have managed to stagger through a computer engineering career with nothing higher than the calculus -- which only served me once, and that indirectly. On the brighter side, I'm generally able to comprehend intelligent explanations, often in polynomial time.

As it stands, I find this article to be incomprehensible and without merit here. Wikipedia is a general reference work. For content to be included here, it has to pass the Joe test: If you had unlimited time in which to explain it (in far more detail than here) to Joe, the Everyman; if Joe was completely cooperative, intelligent, and patient; if eventually he understood all you intend, then could he imagine any possible way in which this subject might be of interest? This topic fails the Joe test.

This article bears a disturbing resemblance to Graveler. It is an isolated swatch of factoplasm lifted out of a highly specialized context, meaningless outside that context. Nothing has been done to show that it has any application to the Real World. While the subject may be a vital part of its own bubble universe, nothing in the article connects it to anything outside.

To be sure, this is longer than Graveler, and arguably factual; but I think that by the time you get past definitions of terms, it is equally valid to say that it is very like a playing card in a mathematical game. If there is a connection to anything concrete, that needs to be shown.

So, I suggest the article be downsized to a bare description of the function and merged into its parent article. (It does at least have some relationship to a larger mathematical topic, does it not?)

Another possibility is to open a new WikiBook entitled "Higher Mathematics" and expand this article into a whole chapter -- there.

I hold out one other route for improvement. I almost began to see a glimmer of light in Applications. Perhaps a diligent effort could actually find some application of this function to something tangible. If so, rewrite this section and move it to the introduction. MBAs and fools like me can read the 4 or 5 sentences that place the function in context, and then we can graze on.

— Xiong熊talk* 05:11, 2005 September 12 (UTC)

So, removing the rhetoric, your proposal for making this article more accessible seems to be to make it smaller or to remove it from Wikipedia completely. I don't see how either of these routes will improve Wikipedia. Gandalf61 11:39, September 12, 2005 (UTC)

To paraphrase, Xiong seems to be saying "I'm the product of a broken social/cultural system, and have no education. I am self-taught, having learned engineering, but otherwise, I'm ignorant. I want to stay ignorant, and everyone else should be kept ignorant too". This sounds nothing so much like the Chinese cultural revolution, where villagers melted down the strong steel of mathematical theorems into puddles of worthless self-taught folk knowledge, and starved by the millions as a result. Shall we picture the "MBA's and fools like him" holding machine guns, eating 4 or 5 rice cakes "in context", and then grazing on? This is a tragedy. linas 14:36, 12 September 2005 (UTC)

Aside from psychoanalysis of this person (is he (or she) really Chinese), he has some point. But he is questioning how wikipedia should be written, and this is probably not a place to do. There are many, too many, hopelessly technical pages already. But the consensus is we should try to revise it so that laymen can understand it, not that eliminate or downsize it from wikipedia. -- Taku 03:31, 14 September 2005 (UTC)

Agree with Taku. And I will say again: please, when you notice a page is too technical, don't just slap a template and walk away. First try to read carefully the page, and see if your criticism is justified. If yes, write at least something on the talk page explaining what you wish were better. That is, be constructive. Oleg Alexandrov 03:57, 14 September 2005 (UTC)

Guess what..the Zeta function IS technical! Maybe "Joe Test" should read a damn book about number theory.

And no matter how uninteresting "Joe" might find this, im sure he's interested by the fact that the Riemann Hypothesis one of the Millenium Problems. As for being useless, this problem is closely related to the primes. Both the distribution of primes and primality testing are two of the biggest problems in number theory, both of which are closely related to cyptanalysis, which is certainly useful. -- He Who Is^{[ Talk ]} 01:58, 29 June 2006 (UTC)

a) This article does not fail the "Joe test" as characterized above. b) Even if it did, the "Joe test" does not reflect Wikipedia policy or intent. -- 98.108.202.144 (talk) 01:56, 31 August 2010 (UTC)

Incomprehensibly Technical

I have read, reread, and then reread this article. I am not a mathematician, but I am also not ignorant of higher mathematical concepts. This article is so technically oriented that it is virtually impossible to comprehend for a layman. In fact, I would wager that unless one already understood the Riemann zeta function that this article would be so complex that it would be useless. Please don't attack me for saying this, I only wish to improve an article that others have obviously worked very hard on. I hope someone will take up the challenge. —Preceding unsigned comment added by 70.121.7.89 (talk • contribs)

I'm a layman who does not find it impossible to comprehend; not even close. You lose your wager. -- 98.108.202.144 (talk) 02:09, 31 August 2010 (UTC)

Unfortunately pessimistic view

You may be right that it's impossible for non-mathematicians to understand most of it. But that may perhaps be true of nearly any article that could be written on this sort of topic.
But I think you're wrong to say that only those who already understand the zeta function can understand this article. I think most mathematicians not familiar with the zeta function would understand it. Michael Hardy 01:30, 28 January 2007 (UTC)

I would wager that unless one already understood the Riemann zeta function that this article would be so complex that it would be useless

Looking at it now, I'd say the "Definition" section, the "Relationship to primes numbers" section, and the "Specific values" subsection would be readily understood by non-mathematicians (even if not by those who simply dislike math and never study math at any level). So I think you're being a bit alarmist. Certainly there are some things here that few besides mathematicians will understand, but far from everything. Michael Hardy 23:20, 4 February 2007 (UTC)

Also there are links to other key concepts, and you might have to do some "stack-based" learning on Wikipedia to really understand it. Still there are some unclear points.67.185.99.246 08:13, 10 February 2007 (UTC)

Euler product formula and probability

This paragraph is not formal at all, and it surely quite confusing. Who says that being coprime with a prime and with another one are mutually indipendent? This kind of trick doesn't always work and often one has to introduce a correction factor. Also, what does "s randomly selected integers" mean? What kind of probabilty distribution are we using on the set of natuaral numbers?--Sandrobt (talk) 16:17, 7 December 2010 (UTC)

• No one says that the paragraph is formal. The result is formal (it can be rigorously proved), but the intuitive explanation leading to it is not.
• There is no probability distribution on all natural numbers being used. As the link explains, the result is about asymptotic probability, i.e., the limit as n goes to infinity of the probability in the uniform distribution over {0,...,n}.—Emil J. 16:53, 7 December 2010 (UTC)

Ok, you're right, I read too quickly and didn't realize you subsituted probability with asymptotic probability. Sorry.--Sandrobt (talk) 02:31, 8 December 2010 (UTC)

42 , riemann zeta & quantum mechanics

please insert this astonishing new informations from Marcus du Sautoy ("The Music of the Primes"): "the moments of the Riemann zeta function...We have known since the 1920s that the first two numbers are 1 and 2, but it wasn’t until a few years ago that mathematicians conjectured that the third number in the sequence may be 42 ... Keating and Snaith not only explained why the answer to life, the universe and the third moment of the Riemann zeta function should be 42, but also provided a formula to predict all the numbers in the sequence"

• http://seedmagazine.com/content/article/prime_numbers_get_hitched/
• http://www.ams.org/notices/200303/fea-conrey-web.pdf (Moments of Zeta, page 349)

188.46.217.137 (talk) 21:13, 12 December 2010 (UTC)

Is this worth noting? or is this a crank?

Stumbled upon this article from 97, http://arxiv.org/abs/physics/9705021 claiming a new formula for \zeta(2n+1). Money is on it being a crank, however, if its actually right then it seemed like something to throw on the page. --Differentiablef (talk) 16:14, 13 April 2011 (UTC)

It's an interesting paper, but not enough to be added in the article (imho). The licterature on the Riemann zeta function is very large and of course the article can't contain everything.--Sandrobt (talk) 20:05, 13 April 2011 (UTC)

That formula for zeta(2*n+1) seems interesting since i don't think anyone was able to 'translate' it to any of the other formulas we know... and i don't mean his definition of B(s) := -s*zeta(1-s) and B'(s)=-zeta(1-s)+s*zeta'(1-s) and then zeta(2*n+1) in terms of B'(s) - since that is just a reformatting of the functional equation of zeta and the derivative (of the functional equation). I mean the formula for B(s) in terms of how Woon calls it 'analytic continuation of an operator'. The closest _looking_ formla to Woons formula for B(s) that we have is i think this http://en.wikipedia.org/wiki/Hurwitz_zeta_function#Series_representation and http://en.wikipedia.org/wiki/Bernoulli_polynomials#Another_explicit_formula. They look very similar to Woons - but still slightly different. But the common wisdom is that there are no _easy_ formulas for zeta(2*n+1) - there are definitly ones we can discount because they are not easy - so maybe that can also be applied to Woons formula. But for example what about zeta(n) = C_n*int(B_n(x)*cot(pi*x),x,0,1) for odd integer n? Here B_n(x) is the nth bernoulli polynomial, cot(x) = cos(x)/sin(x), int(f(x),x,0,1) is the integral for x over [0,1] and C_n is expressible in powers of pi and factorial(n) (see further 'Abramowitz and Stegun: Handbook of Mathematical Functions' http://people.math.sfu.ca/~cbm/aands/page_807.htm). Is that worth mentioning? — Preceding unsigned comment added by Petersheldrick (talk • contribs) 00:09, 8 July 2011 (UTC)

Analytical Continuation (yet again)

Without wishing to enter into a debate over the modern convention for defining the function, I would like to point out that Riemann himself defined it as a contour integral, not by analytical continuation. It is evident from Riemann's paper that this is so. There is quite a good discussion of this point also in section 1.4 of Edwards' book, which you include in the References section. — Preceding unsigned comment added by 58.168.69.120 (talk) 00:22, 13 October 2011 (UTC)

Ambiguity in caption on first diagram?

The first diagram presents a color-based encoding of the values of ζ(s) in the complex plane. The last sentence of the caption to that diagram says: "Positive real values are presented in red." I'm wondering if that should more precisely read: "Positive real parts are presented in red", especially as the word "value" is used previously to refer to the value of ζ(s), not to the [real] value of its real or imaginary part. — Preceding unsigned comment added by 210.9.140.245 (talk) 09:53, 18 March 2012 (UTC)

Nope, the picture shows the values of ζ(s); red areas refer to real values, not its real parts. Nageh (talk) 01:10, 19 March 2012 (UTC)

So all values of the zeta function in the entire half-plane Re(s) > 1 are real and positive? Equivalently, the Dirichlet series of which the zeta function is the analytic continuation converges to a positive real number for any s satisfying Re(s) > 1? — Preceding unsigned comment added by 122.110.54.252 (talk) 03:43, 19 March 2012 (UTC)

Is there some mathematician who can clear up my confusion? If I'm reading the caption correctly, the red expanse in the first diagram in the article indicates that ζ(s) is everywhere real and positive in the half-plane Re(s) > 1. The function obviously takes real values for real s > 1 but is it actually real and positive elsewhere in that half-plane? Or am I misunderstanding the diagram? — Preceding unsigned comment added by 210.9.140.245 (talk) 20:21, 21 March 2012 (UTC)

Still no answer to the previous question but in the meantime I've made a slight correction to the series definition of ζ(s) in the Definition section, where it had a set membership sign (read as "σ belongs to the real part of s > 1") that was obviously meant to be an equals sign. — Preceding unsigned comment added by 210.9.140.245 (talk) 07:45, 6 April 2012 (UTC)

That color shows the argument of the function, to an argument close to 0 correspond a color close to red (or, more precisely, close to the red color corresponding to arg=0), i.e. to a value close to a real correspond a color close to red. On the half plane with real part >1 the first summand (1^-s=1) in the definition of the Riemann zeta function is dominant and the function si very close to 1 and so to a real and therefore it has a red color on the graph. --Sandrobt (talk) 22:11, 8 April 2012 (UTC)

Thanks for your response but I'm still confused. I think the caption needs to be rewritten for clarity. The relationship between color and hue is quite unclear, and I can't tell whether they're meant to be independent attributes of each pixel. There seems to be a "parent" attribute (color) with two child attributes, "dark color" and "hue", which makes no logical sense to me.

1. "dark colors denote values close to zero": This seems OK at first glance, and consistent with zeros being colored black, but how does darkness relate to hue and to color? If color is one of the encoding mechanisms, I would expect it to be defined without any adjective like "dark" in front of it. Should I think of the color as encoding the modulus?
2. "hue encodes the value's argument": My confusion continues here. Is "hue" synonymous with "color"? (It can be in colloquial usage.) Or does it mean intensity, i.e., degree of darkness, which is closer to its primary definition?
3. And finally, "Positive real values are encoded in red": Is this a definition or merely an example of whatever encoding was previously defined? And if the "values" referred to here are values of the argument [of the values of the function], then surely the word "real" is redundant.

If there are two independent attributes of each pixel that constitute the encoding, then the significance of each needs to be clearly defined, and with no interaction between them. If I knew enough about the RZF I could probably work out what the caption is trying to say and could redraft it myself. But I don't, so I can't. As someone who understands the rudiments of complex-valued functions of a complex variable (which is all the knowledge that the diagram presupposes), I find the caption in its current form very confusing. I appreciate that my difficulty may not be comprehensible to someone with a good understanding of the RZF but I presume that the article, and certainly that diagram, is written for the likes of myself, not for RZF specialists. — Preceding unsigned comment added by 210.9.140.245 (talk) 10:18, 9 April 2012 (UTC)

I understand you are not familiar with the HSL (hue-saturation-lightness) color model. Have a look. Nageh (talk) 13:04, 9 April 2012 (UTC)

I wasn't and thank you. If I'm understanding things correctly, the diagram uses two of the three HSL attributes, viz., hue (to represent the argument) and lightness (to represent distance from zero, or equivalently the modulus).

Actually I don't think it's either necessary or desirable to require or assume familiarity of the reader with the HSL model, or to have to point the reader to any explanation of it. In any case, since hue is referred to explicitly (with a link to its Wikipedia entry) I would expect the same for lightness, which as far as I can tell is referred to only implicitly (in the phrase "dark colors"). The fact is, the information that needs to be conveyed in the caption is simple enough that informal English can be used for greater clarity and without any loss of precision. Thus, I would suggest use "color" and avoid "hue" entirely, and perhaps use "intensity" to capture the lightness attribute. In that way, the caption would be immediately intelligible to any reader regardless of their familiarity with HSL and without the reader having to wonder about the difference between "color" and "hue", or indeed whether there is any (which was another source of my confusion).

I'm still a little confused by the statement "Positive real values are presented in red", which doesn't seem to sit well with either the previous use of the term "values" ("dark colors denote values [of the function] close to zero") or the HSL model. If those values are of the argument (as previously explained by sandrobt) it seems a weird thing to say, as arguments are always real anyway. This was the source of my original confusion. I would therefore suggest rewriting the last sentence of the caption to avoid the word "real" and to make it clear that red represents values of the function that have a positive argument. I find the use of the unqualified word "values" in the last sentence ambiguous and confusing.

Finally, a question: At the scale at which the diagram is drawn, would no significant variation in the lightness of the red expanse be visible? In other words, over the red-colored half-plane does the function take only values whose (close) distance from zero can't be visually distinguished on the scale of the diagram? I'm assuming that the red is meant to be a "dark" version of that color (= hue) and thereby denotes proximity to zero as per the caption ("dark colors denote values close to zero").

This may all sound somewhat pedantic but I'm speaking from the point of view of the audience to whom I believe the article is addressed, viz., someone literate in English and partially literate in complex functions. It's a nice diagram, which uses a powerful visual technique to capture the reader's attention and illustrate some salient aspects of the behavior of the function. I think it's therefore reasonable to expect a comparable level of clarity in the wording of the caption as in the mathematical discussion that it accompanies.

Yes, you are still confused. :) As you correctly inferred, a "color" has two attributes: "lightness", encoding a value's modulus, and "hue", denoting its argument. However, I'm still not sure whether you understand what "dark" color and [low] "lightness" means: it means colors close to black, not close to saturation (e.g., fully red, fully blue, etc.). Now apply that understanding to the sentence "Positive real values are presented in red". It simply means that values that are positive reals (i.e., having an argument that is zero) are encoded in red color. However, that sentence was potentially misleading in the context, and I have updated it. See if it makes more sense now for you.

Indeed, values on the right half plane have quite small moduli, but they are not too close to zero: if they were, they would be colored almost black. In contrast, if the moduli were large they would be encoded as near-white colors.

Now regarding your suggestion of using different vocabulary. I agree that we cannot assume every reader's familiarity with the HSL or HSV color model. Unfortunately, the term "color" is more ambiguous than you'd think, and so replacing "hue" by color simply doesn't work. "Intensity", as you suggest, is actually either radiance or saturation, so that's even worse. The next-best I could think of was replacing hue by spectral color, though that is not accurate. In fact, the best I could think of was replacing "dark color" by "colors close to black", and linking hue to its article, as I did with the recent edits.

Hope that clarifies things a bit. :) Nageh (talk) 12:29, 11 April 2012 (UTC)

Thanks for the further clarification. So it's true that positive real values are encoded in red — as well as a lot of other values! I was mistakenly (though I think understandably) reading the sentence as a definition of "red" and thus inferring the unintended converse of your statement. Your new wording is much better. I would suggest one more improvement for maximum clarity: "Values with arguments close to zero (in particular, positive real values on the real half-line) are encoded in red". This just makes it absolutely clear that the red encoding applies to positive reals by virtue of a more inclusive encoding criterion, and also tells the reader where those positive real values actually occur among that vast expanse of red. — Preceding unsigned comment added by 210.9.140.245 (talk) 21:06, 11 April 2012 (UTC)

I have rephrased it a bit. It seems a bit wordy but what gives. Hope that is satisfying now. Nageh (talk) 23:55, 12 April 2012 (UTC)

It's clear now, thank you. — Preceding unsigned comment added by 210.9.140.245 (talk) 09:49, 14 April 2012 (UTC)

That diagram is not explained very well ... where is the "white spot" at s1. First of all, why isn't "S" labelled on the axis. Which Axis? If you label an axis "S" then we could find S1 but you didn't and then I don't see any "white spot" anyways. So, if you could label which axis is S and then maybe make the "white spot" maybe bigger so we can actually see it because I don't see a white spot do you?

Ty

173.238.43.211 (talk) 18:53, 6 May 2012 (UTC)

There is no axis for s in the common sense because s denotes a complex number in the plane. Maybe the axes could be labeled "Re s" and "Im s". But anyway, it shouldn't take a lot of ingenuity to figure out where s = 1 is. Nageh (talk) 19:16, 6 May 2012 (UTC)

Ya well it shouldn't take a lot of ingenuity to figure out THERES NO WHITE SPOT ON THAT GRAPH. Its bad enough you think a reader can just assume which axis you are talking about but its even worse that you claim theres a "white spot" and then not even label which axis its on. I can only assume that kinda black/green/white "bubble" area in the middle of the chart is supposed to be a "white spot" but it happens to have black and green in it also so no idea where this "white spot" is or which is s1 so does anyone know who made this graph so he can explain it so we get it? Do you mean the black egg shaped spot in the centre?? Is that black spot the "white spot"?

173.238.43.211 (talk) 08:24, 7 May 2012 (UTC)

Sigh. (1) I assume nothing. (2) I didn't create that graph. (3) There is a white spot right at s = 1 as claimed. (4) The spot is not along an axis but in the plane. As you cannot find it I can only assume you have never before seen an image drawn in the complex plane. There is a scale at the bottom of the image. Go slightly to the right of zero to be approximately at position one, then draw a vertical line up the image. Along this line you will see all the stripes coming from the left fading out. All of them end in a black spot, except for the middle one, which ends in a white spot. It's the right-most area of the "bubble" you mentioned. (5) Anyone can contribute to Wikipedia. If you feel you can provide a better image please go ahead. Nageh (talk) 11:52, 7 May 2012 (UTC)

Ok, thank-you. Perhaps someone else that doesn't see the "obvious white spot on the unmarked axis consisting of blue/green/ and yellow colors" will hopefully read through this talk page to find out that the blue/green/yellow spot is the "white spot". Do you see how small that "white spot" is? Its a pin prick. If only the supposed "white spot" area was circled then noone would ever question where this "white spot" that has blue, green, and yellow in it actually is. That was only the point of me asking where it was. It was where I kind of guessed it is but wasn't sure since the diagram in no way makes it easy to know for sure was all I wanted to get across. Sorry if my question bothered you. We all win in the end.

173.238.43.211 (talk) 18:19, 9 May 2012 (UTC)

Oh and this article and graph about the zeta function is less problematic to understand in my opinion;

http://simple.wikipedia.org/wiki/Riemann_hypothesis

173.238.43.211 (talk) 12:31, 10 May 2012 (UTC)

Yeah, especially because it does not show the singularity at s = 1. But whatever. Nageh (talk) 13:33, 10 May 2012 (UTC)

Maybe calling it a "singularity" instead of "white spot" at s1 would be better since its really blue/green/ and yellow and the axis isnt labelled so we could know for sure - just my opinion

173.238.43.211 (talk) 22:39, 10 May 2012 (UTC)

Möbius function

The link to Möbius function leads to a function defined only for integers. I suspect that this is not the function intended.Klausok (talk) 08:13, 23 October 2012 (UTC)

Yes, it is.—Emil J. 11:35, 23 October 2012 (UTC)

Klausok is obviously not familiar with arithmetical functions. — Preceding unsigned comment added by 92.28.8.213 (talk) 16:09, 26 October 2012 (UTC)

Hi, I think this should be mentioned at the original page

Log[2/3/E^(5 (Pi/2)) + E^(7 Pi)/(7/2/E^(7 (Pi/2)) + 5/2/E^(5 (Pi/2)) + 3/2/E^(3 (Pi/2)) + E^(5 (Pi/2)) + 2 Pi)] = 14.134725141734373744769652011837891376454997031685134...

But the first zero on the critical line is Zeta(1)=

 14.134725141734693790457251983562470270784257115699...

Have a nice day, Dietmar — Preceding unsigned comment added by 217.81.127.212 (talk) 16:08, 5 November 2012 (UTC)

I have no idea what is the Log[...] expression doing in the article in the first place. There are much simpler approximations of the first zero to 12 digits of accuracy (such as its continued fraction approximants), and there is no explanation why this complicated formula should have any deeper meaning. I’ll remove it.—Emil J. 18:36, 5 November 2012 (UTC)

Zeta (1/2)

How is the zeta function of a half calculated? In fact, how is the zeta function of any number between zero and one calculated?! — Preceding unsigned comment added by 86.136.129.187 (talk) 19:12, 3 December 2012 (UTC)

The zeta function is defined for Re(s) < 1 by analytic continuation. As for actual computation, some of the expressions in the section Riemann zeta function#Representations can be used.—Emil J. 14:29, 4 February 2013 (UTC)

Offer to re-write

Would it be possible for me to re-write the article from scratch, and then to go from there?

The article is in a deplorable state. It is too technical and in addition the technical parts are not central to the current (or past) research on $\zeta(s)$. The emphasis on "Selberg's conjecture" (not a conjecture in the first place but more of a remark in Selberg's paper) is badly misleading.

It seems that somebody very fond of Karatsuba keeps putting these here. It is not normal to have 8 references to Karatsuba and 1 to Shanker (never heard of him) and no references to Hardy-Littlewood, Ingham, Bohr-Landau, Titchmarsh and more recently to for example Conrey, Soundararajan, Iwaniec, etc.

Anyway, if this is not possible then I will go away and the article will remain in its deplorable state... — Preceding unsigned comment added by Karatsuba (talk • contribs) 07:50, 9 June 2013 (UTC)

Basically everything after and including the section "Zeros, the critical line, ..." needs to be purged and re-written. There is too much misleading emphasis (i.e Karatsuba, identities which nobody uses) there and too much original research (i.e Shanker).

Ramachandra -> Ramanujan

The Section "Estimates of the maximum of the modulus of the zeta function" contains a typo: "The case $H\gg \ln \ln T$ was studied by Ramachandra" should read "The case $H\gg \ln \ln T$ was studied by Ramanujan". I'd not go as far as editing the article as I would not dare to touch an article on mighty maths, just a suggestion. — Preceding unsigned comment added by 152.66.244.111 (talk • contribs) 15:13, 13 May 2011 (UTC)

No the result is due to Ramachandra and not Ramanujan. — Preceding unsigned comment added by Karatsuba (talk • contribs) 07:56, 9 June 2013 (UTC)

Recent removals

Recently text on the so-called Selberg conjecture was removed by an anonymous IP editor. This removal was reverted (on somewhat superficial grounds). It seems to me that the IP has a legitimate point. The material under discussion at best seems like undue weight, and at worst original research. (See the comments also in the preceding section, which pertain to the same material). May I suggest that we leave this material out unless the case for its inclusion is made more clear. Sławomir Biały (talk) 01:55, 11 June 2013 (UTC)

Zero density theorems

There should be more on zero density theorems here or in the article on the Riemann hypothesis. — Preceding unsigned comment added by 86.181.154.200 (talk) 12:55, 28 December 2013 (UTC)

On the recent revert removing the Brady Haran video

The recent addition of an external link to Brady Haran's video on the zeta function was reverted as Added link does not add any new information about the subject (just mentions it). The subject of the video was "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12". This brings up three points.

1. If you learn visually, having a video that explains Specific Value #4 is beyond useful, particularly given the state of this article as discussed above.
2. The associated remark to #4 "which is only true in a very informal sense (i.e. this is just an abuse of notation, and the series does not actually converge to this value)" is an insult to Euler, who proved it that it does so converge, as Dr. Betz remarks in the external links to 1 + 2 + 3 + 4 + · · ·.
3. 1 + 2 + 3 + 4 + · · · itself says "the analytic continuation of ζ(s) gives ζ(−1) as −1/12."

I'm not interested in an edit war, so I'll neither put the link back nor remove the deminishment of Euler. But somebody should. user:JMOprof ©¿©¬ 18:22, 10 January 2014 (UTC)

Zeta(-1)

There seems to be an edit war going on at the moment, and we should resolve it here on the talk page rather than by a bunch of different editors reverting and re-reverting.

Is Zeta(-1) = 1 + 2 + 3 + 4 .... ? Until we can come to consensus and reliably source this as settled, we probably shouldn't put it in the article, or if we do we need to be careful to provide context so that it is not misunderstood by the lay reader. My understanding of wiki policy is to leave it out until a consensus is reached.

A couple of observations:

• The article 1 + 2 + 3 + 4 + · · · looks pretty good and seems to treat it carefully. We should be consistant across both articles.
• I'm not sure whether Quora should be accepted as a reliable source.
• If it is, why are we citing the answer written by a "former PhD candidate" instead of one written by a full professor who basically claims the opposite?

There's a quite misleading video making the rounds, and many users are coming here to "verify" whether it's true or not. We need to provide a solid article. Looking forward to comments. Mr. Swordfish (talk) 19:03, 21 January 2014 (UTC)

Short answer: no, zeta(-1) is not equal to this divergent series. The Dirichlet series representation of zeta(s) is only valid for Re(s) ≥ 1. The series 1+2+... has unbounded and strictly increasing partial sums: it diverges properly to +infinity. There might be situations where it is convenient to mean by 1+2+... something other than the usual meaning of a series. For instance, if one thinks of the Dixmier trace of certain operators that aren't properly trace class. The trace can still in many cases be defined using spectral theory and analytic continuation. It's in precisely this context that zeta(-1) appears in string theory (aside: I don't really like how issues like this are dealt with by waving the magic wand of string theory in lieu of saying anything substantive). Although in these cases, we often think of the trace as the sum of the eigenvalues, it's not actually true. In fact, it would make a lot of mathematics substantially easier if it were actually true. Sławomir Biały (talk) 19:38, 21 January 2014 (UTC)

I definitely prefer this version (per a recent revert by Sławomir Biały) to its predecessor. We should reference the divergent series 1+2+3+... without saying that zeta(–1) is that series. This version strikes the right balance. And for a subject about which we have so many expert sources to draw on, there is no need to fall back to less-reliable sources such as Quora. —David Eppstein (talk) 20:20, 21 January 2014 (UTC)

My take is that saying 1 + 2 + 3 + 4... = -1/12 is about like saying that 7 + 5 = 1. Now, this second statement is actually true in Z₁₁ , but stating it without providing the context of modulo arithmetic or quotient rings would be misleading. The notions of "+" and "=" in Z₁₁ are different than what most people are used to. Likewise, Zeta(-1) can be expressed formally as 1+2+3+4... but linking the two with an equals sign is using "=" in a way that is different than conventional usage (or perhaps, as suggested above, it is 1+2+3... that is being used as something other than its usual meaning). I've no problem with doing that in the article as long as the proper context is provided, but my editorial judgment says that trying to do so may cause more confusion than it clears up. I support the revision by Sławomir Biały - Mr. Swordfish (talk) 15:54, 22 January 2014 (UTC)

As of today, the formula

$\zeta (-1)=1+2+3+4+\cdots =-{\frac {1}{12}}$

is back in the article. Are we going to just keep removing and restoring it on a weekly basis, or are we going to try to come to some consensus here on the talk page? Let's hear the reasons for including it. Mr. Swordfish (talk) 16:33, 17 February 2014 (UTC)

This article and the article 1 + 2 + 3 + 4 + · · · should agree, and unless we have a news flash here, 1 + 2 + 3 + 4 + · · · says that Euler proved 1 + 2 + 3 + 4... = -1/12. I'm inclined to take Euler's word for it myself. See Drs. Baez's and Cais's articles in the external links to 1 + 2 + 3 + 4 + · · ·. Dr. Cais says

Such a statement obviously presents philosophical difficulties. Namely, one is forced to ask how the “sum” of a divergent series of entirely positive terms can be negative. Yet the manipulations involved in our determination of s are no more outlandish than those used in determining 1 − 1 + 1 − 1 + ··· = 1/2. We will see later that in a very precise sense, −1/12 is the correct value of 1 + 2 + 3 + 4 +···.

—Dr. Bryden Cais

user:JMOprof ©¿©¬ 17:10, 17 February 2014 (UTC)

Euler's proofs are not considered to be sufficiently rigorous by modern mathematics. —David Eppstein (talk) 17:19, 17 February 2014 (UTC)

This article and the article 1 + 2 + 3 + 4 + · · · should agree...

No argument there, but that article appears to be experiencing the same edit war as this one. Perhaps that talk page would be a better venue than this one, but we really need to come to consensus on the talk pages rather than an endless stream of edits, reverts, re-reverts, etc. Mr. Swordfish (talk) 19:37, 17 February 2014 (UTC)

Auto archive?

With 64 threads, some of which are approaching their second decade, how about implementing auto archive? I will volunteer to do it if there is consensus. Mr. Swordfish (talk) 16:55, 18 February 2014 (UTC)

I agree. Take the action needed, Mr swordfish. — Preceding unsigned comment added by 90.208.184.124 (talk) 14:59, 21 March 2014 (UTC)

Seeing no objections, auto-archive has been implemented. Mr. Swordfish (talk) 11:27, 25 March 2014 (UTC)

Need clarification on where on the complex plane an expression is valid

You say that the Riemann zeta function can also be defined by the integral:

$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\mathrm {d} x$

But you never stated whether this definition is valid everywhere on the complex plane (wherever the function exists). People just learning about Zeta function regularization of what I like to call the Ramanujan (1+2+3+...) are interested in the analytic continuation of the Riemann zeta function over the entire complex plane. It seems trivial to me that the aforementioned definition is valid over the entire plane, but I might be wrong. You need to either state that this equation is valid for the entire complex plane, or you need to help people like me who will look at the equation, do a bit of sloppy mental math and conclude (falsely?) that the expression does define the function everywhere that it exists.--guyvan52 (talk) 15:18, 21 December 2014 (UTC)

If it the integral expression is valid over the complex plane then perhaps this:

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. (The notation with s, σ, and t is traditionally used in the study of the ζ-function, following Riemann.)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

$\zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots \;\;\;\;\;\;\;\sigma ={\mathfrak {R}}(s)>1.\!$

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series. Eqivalently^reference?, it can also be defined everywhere that it exists on the complex plane by the integral

$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\mathrm {d} x$

--guyvan52 (talk) 15:18, 21 December 2014 (UTC)

Vague

Under Various properties and Reciprocal is the statement "the claim that this expression is valid". This expression? Which one? John W. Nicholson (talk) 04:54, 15 March 2015 (UTC)

I would assume "this expression is valid" means that the Dirichlet series converges. —David Eppstein (talk) 05:34, 15 March 2015 (UTC)

Assume? And, what is going to be done to fix it? John W. Nicholson (talk) 14:50, 15 March 2015 (UTC)

recent change

Someone changed the equation $\zeta (s)={\frac {2^{s-1}}{s-1}}-2^{s}\!\int _{0}^{\infty }\!\!\!{\frac {\sin(s\arctan t)}{(1+t^{2})^{s/2}(e^{\pi \,t}-1)}}\,\mathrm {d} t,$ into: $\zeta (s)={\frac {2^{s-1}}{s-1}}-2^{s}\!\int _{0}^{\infty }\!\!\!{\frac {\sin(s\arctan t)}{(1+t^{2})^{s/2}(e^{\pi \,t}+1)}}\,\mathrm {d} t,$ (i.e.: minus 1 changed into plus 1 in the denominator)

This seems suspicious, since there is no explanation, and no other edits from that IP. Can someone looks into it to verify if it is legitimate? — Preceding unsigned comment added by Dhrm77 (talk • contribs) 05:20, 25 March 2016 (UTC)

Confusing passage about universality

This paragraph first states "it was first proved by blah in 19xx" but then says "Nevertheless, none of the proofs above are definitive or complete, since a definitive proof would imply that Riemann's Hypothesis is true."

Is it proved or not proved? Whats missing? This is confusing to me. — Preceding unsigned comment added by 32.215.35.33 (talk) 18:54, 21 June 2016 (UTC)

Well yes, it is not written exactly clearly. But idea is, that someone proved something while considering that Riemann Hypothesis is true, so basically if Riemann Hypothesis is true then whatever that someone proved will be also true, but if it is not, then it might not be. So basically they proved something, but took an unproven hypothesis as basis for their proof. Trimutius (talk) 15:40, 22 June 2016 (UTC)

That whole section from that point down starts assuming that the reader knows things that have not yet been explained or defined. I would say the entire page needs to be reworked, starting at the top with definitions. 71.48.255.210 (talk) 20:56, 14 July 2016 (UTC)

is n a zero of the fonction if and only if ζ(n)=0 ?

i understand nothing to this notion of «ζ(x) zero», so if true, i think we should ad it to explain — Preceding unsigned comment added by BeKowz (talk • contribs) 12:59, 3 December 2016 (UTC)

Relationship Between $\log \zeta (s)$ and $\pi (x)$

The article mentions the following relationship between $\log \zeta (s)$ and $\pi (s)$ under the heading "Mellin transform".

$\log \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\ dx,\quad Re(s)>1$

I was hoping to explore the relationship above using the Fourier series representation for $\pi (x)$ which is an infinite series of Fourier series where each Fourier series consists of an infinite series of Sine terms (see Fourier Series Representations of Prime Counting Functions). Unfortunately a term of the form ${\frac {sin(a\ x)}{x(x^{s}-1)}}$ resists integration. However, I have been able to explore the relationship above using the relationship below was derived from the original relationship above via integration by parts.

$\log \zeta (s)=\int _{0}^{\infty }\pi '(x)(s\log(x)-\log(1-x^{s}))\,dx$

I haven't been able to explore the second relationship above using the Fourier series representation for $\pi '(x)$ because a term of the form $cos(a\ x)\log(1-x^{s})$ also resists integration, but the second relationship above can be evaluated by writing the integral as a sum as follows.

$\log \zeta (s)=\sum _{p\in Primes}(s\log(p)-\log(1-p^{s}))$

Using the sum formula for the second relationship above, I confirmed the the real part of $\log \zeta (s)$ converges to some degree for the two relationships above for $Re[s]>1$, but the imaginary part of $\log \zeta (s)$ diverges significantly using the sum formula above which makes me suspect the validity of the two relationships above.

StvC (talk) 01:04, 5 November 2016 (UTC) StvC (talk) 00:49, 15 December 2016 (UTC) StvC (talk) 05:15, 16 December 2016 (UTC) StvC (talk) 16:46, 16 December 2016 (UTC)

Gamma Function definition

In the Riemann zeta function page it is written that:

$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {1}{e^{x}-1}}\,x^{s}{\frac {\mathrm {d} x}{x}}$

and that:

$\Gamma (s)=\int _{0}^{\infty }e^{-x}\,x^{s}{\frac {\mathrm {d} x}{x}}$.

However looking at the Gamma function definition in Wikipedia one can see that:

$\Gamma (s)=\int _{0}^{\infty }e^{-x}\,x^{s-1}{\frac {\mathrm {d} x}{x}}$.

Correspondingly $\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {1}{e^{x}-1}}\,x^{s-1}{\frac {\mathrm {d} x}{x}}$.

More looking at Riemann's paper he has the s-1 also.

However Riemann instead of Gamma used factorial $\Gamma (n)=(n-1)!=\Pi (n-1)$.

I tried to fix the page but it was rejected as vandalism.

Please correct it or explain to me what am I wrong about.

Adikatz (talk) 06:52, 26 July 2017 (UTC)

In the definition of the zeta function the definition of the Gamma function is wrong. It should be s-1 instead of s. The s-1 should also be in the integral representing the gamma function. I have tried to fix but it was revoked as vandalism.

Vandalism is "malicious or ignorant destruction" (Webster's). In your case the second definition occurs. The gamma function is defined as

$\Gamma (s)=\int _{0}^{\infty }e^{-x}\,x^{s-1}{\mathrm {d} x}$. Or $\Gamma (s)=\int _{0}^{\infty }e^{-x}\,x^{s}{\frac {\mathrm {d} x}{x}}$. But not $\Gamma (s)=\int _{0}^{\infty }e^{-x}\,x^{s-1}{\frac {\mathrm {d} x}{x}}$.Sapphorain (talk) 07:57, 26 July 2017 (UTC)

Sorry. I was mistaken since I did not pay attention to the fact that the dx was divided by x. Adikatz (talk) 14:21, 26 July 2017 (UTC)

Some more representations

Here are some more representations, but I don't know what subsection to put them in, or what commentary to give on them:

$\zeta (n)=\underbrace {\int _{0}^{1}\int _{0}^{1}\ldots \int _{0}^{1}} _{n}\,{\frac {\mathrm {d} x_{1}\mathrm {d} x_{2}\ldots \mathrm {d} x_{n}}{1-x_{1}x_{2}\ldots x_{n}}}$ for positive integer n

$-{\frac {\zeta (s)}{s}}=\int _{0}^{\infty }\operatorname {frac} \left({\frac {1}{t}}\right)t^{s-1}\mathrm {d} t$ for 0 < Re(s) < 1, where frac is the fractional part

$\Gamma (s+2)\zeta (s+2)=\int _{0}^{1}\int _{0}^{1}{\frac {[-\ln(xy)]^{s}}{1-xy}}\,\mathrm {d} x\,\mathrm {d} y$ for Re(s) > 1

(Reference: http://mathworld.wolfram.com/RiemannZetaFunction.html)

--AndreRD (talk) 16:29, 27 July 2017 (UTC)

If s=1/2 in Riemann functional equation then the calculated result seems to disprove Riemann's hypothesis

$\zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s),$

If the complex number s = 1/2 + 0j is put into the Riemann functional equation after the equation is rearranged so the left hand side reads E{s}/E{1-s} then the left hand side with a value of 1 does not equal the right hand side and this means that the complex part is not zero and so there can be no solution on the 1/2,0 coordinate of the complex plane as Riemann says there is. Have I got this right?

Soopdish (talk) 12:48, 8 August 2017 (UTC)Soopdish

The conjecture does not say "every point on the critical line is a zero", it says "every zero is on the critical line". So the fact that the point you looked at is nonzero is uninteresting and irrelevant. —David Eppstein (talk) 16:29, 8 August 2017 (UTC)

Correction Integral Section

The Integral sections shows a formula for the zeta function that is super wrong. The correct formula can be found here Abel–Plana formula
It should look like this: $\zeta (s)={\frac {1}{s-1}}+{\frac {1}{2}}+2\!\int _{0}^{\infty }\!\!\!{\frac {\sin(s\arctan t)}{\left(1+t^{2}\right)^{\frac {s}{2}}\left(e^{2\pi t}-1\right)}}\,\mathrm {d} t$
The provided link #24 is also wrong. It links to an arc length calculus page.
--EmpCarnivore (talk) 21:19, 7 September 2018 (UTC)

Perfect powers

Is there a formula out there that relates the zeta function to the perfect powers (1, 4, 8, 9, 16, 25, 27, 32, etc) in a similar way that Euler's product formula does for primes? The reason I ask is because I discovered a very simple one a while back, but I can't find information about any other such formulas. Thanks. --Vagodin 14:47, August 21, 2005 (UTC)

Do your own research. — Preceding unsigned comment added by 2A00:23C0:FC81:5A01:8515:7665:AFC2:DCC4 (talk) 10:13, 12 September 2018 (UTC)

Globally convergent series

The 'globally convergent series' found by Hasse appears to be essentially just the Euler transform applied to the Dirichlet eta function. — Preceding unsigned comment added by Carifio24 (talk • contribs) 23:22, 4 July 2007 (UTC)

I programmed the second of the two Hasse series, and the second one seems to be wrong. It returns large incorrect values. If there's a bug, I must be blind:

The first series (which works correct) uses the following code:

double hasse0(double s, uint64_t limit) {

 double u = 0;
 for (uint64_t n=0; n<limit; ++n) {
   double r = 0;
   for (uint64_t k=0; k<=n; ++k) {
     double t = choose(n,k) / pow(k+1,s);
     if (k%2) r -= t; else r += t;
   }
   u += r/pow(2,n+1);
 }
 return u/(1-pow(2,1-s));

}

The second series (which does not work) uses the exact same template:

double hasse1(double s, uint64_t limit) {

 double u = 0;
 for (uint64_t n=0; n<limit; ++n) {
   double r = 0;
   for (uint64_t k=0; k<=n; ++k) {
     double t = choose(n,k) / pow(k+1,s-1);
     if (k%2) r -= t; else r += t;
   }
   u += r/(n+1);
 }
 return u/(s-1);

} — Preceding unsigned comment added by Pcp071098 (talk • contribs) 23:07, 27 August 2013 (UTC)

Graph

Under "Specific values", the graph seems to be of three functions, only one of which is the Zeta function. The other two seem to be based on a finite number of terms of the infinite series in the definition of the Zeta function. — Preceding unsigned comment added by 79.79.29.112 (talk) 08:15, 28 July 2017 (UTC)

Simple Approximation

$\zeta (x)\approx {\begin{cases}{\frac {1}{x-1}}+\gamma &{\mbox{if }}1\leq x\leq 2\\\\\pi ^{x}/(3{\sqrt {10}}^{x-1})&{\mbox{if }}x>2\end{cases}}$

Also, for $x>1$ we have:

$\lim _{x\to 1}[\zeta (x)-{\frac {1}{x-1}}]=\gamma$

where, in both expressions, $\gamma$ refers to the Euler-Mascheroni constant. — Craciun Lucian.

Many of these remarks are not new. — Preceding unsigned comment added by 2A00:23C0:FC81:5A01:8515:7665:AFC2:DCC4 (talk) 10:35, 12 September 2018 (UTC)

What is sigma?

In the section on the proof of the functional equation, the text keeps mentioning criteria on sigma, like sigma > 0, sigma > 1. But then the actual computations have no sigma. They're all in terms of s, is sigma meant to be s here? -lethe ^{talk +} 19:04, 29 January 2020 (UTC)

More likely, $\sigma =\operatorname {Re} (s)$. – Tea2min (talk) 19:32, 29 January 2020 (UTC)

I have looked at the source text by Titchmarsh. The parametrization is given on page 1: s = sigma + it. As you said. Thanks. -lethe ^{talk +} 19:47, 29 January 2020 (UTC)

Definition

Why use $\sum _{k=1}^{\infty }{\frac {1}{k^{n}}}$, not $\sum _{k=1}^{\infty }k^{n}$, as $\zeta (n)$? I think $\sum _{k=1}^{\infty }k^{n}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{n}}}$ is more “natural”. Three similar functions are $\sum _{k=1}^{\infty }(-1)^{k-1}k^{n}$, $\sum _{k=1}^{\infty }(2k-1)^{n}$ and $\sum _{k=1}^{\infty }(-1)^{k-1}(2k-1)^{n}$. — Preceding unsigned comment added by Xayahrainie43 (talk • contribs) 13:54, 24 September 2018 (UTC)

The suggestions made by X etc. don't converge. — Preceding unsigned comment added by 2A00:23C4:7C87:4F00:25BC:B3AF:C915:446B (talk) 15:08, 1 September 2020 (UTC)

Anon edit needs vetting

The following anonymous edit comes from an IP with a very checkered history. It needs vetting:

$-\zeta (s)=\prod _{p}{\frac {1}{1-p^{-s}}}+\zeta (s)=\prod _{p\in primes}{\frac {1}{1-p^{-s}}}$

Thanks. --Wetman 13:02, 18 Apr 2005 (UTC)

This is surely wrong, as it implies that the \zeta function is zero. Oleg Alexandrov 18:15, 18 Apr 2005 (UTC)

I don't see this exact thing in the article though. Oleg Alexandrov 18:18, 18 Apr 2005 (UTC)

Wetman posted a diff; he is talking about the edit from 9:32, 8 April 2005, which Oleg reverted a few hours later anyway. No matter, the edit was fine, the product is over primes. Case closed. linas 22:09, 18 Apr 2005 (UTC)

Representations - Dirichlet series

This section seems suspect since 1) the reference [21] is just a reference to the cauchy integral theorem, and 2) the series presented does not appear to be a dirichlet series. Can someone confirm if this is even true? — Preceding unsigned comment added by 208.38.59.163 (talk) 20:19, 23 September 2020 (UTC)

Note: it does give the correct value of zeta at 2, but i am still not seeing where it comes from? — Preceding unsigned comment added by 208.38.59.163 (talk) 20:29, 23 September 2020 (UTC)

Knopp

Under "Globally convergent series", I think that Knopp's conjecture was made in 1926. — Preceding unsigned comment added by 213.48.238.18 (talk) 13:42, 3 November 2020 (UTC)

Clarity needed

Under "Generalizations", the Clausen function is mentioned, ungrammatically. — Preceding unsigned comment added by 79.77.163.188 (talk) 13:01, 26 April 2021 (UTC)

Alleged polar graph

This article uses the picture https://en.wikipedia.org/wiki/Riemann_zeta_function#/media/File:Zeta_polar.svg

Note that it is not a polar graph in any sense, it is a regular Cartesian graph, with $x(t)=\Re \zeta (1/2+it)$ and $y(t)=\Im \zeta (1/2+it)$, $t\in [0,34]$. It should be corrected and renamed; the current version is misleading. A1E6 (talk) 18:17, 21 July 2021 (UTC)

Do we know that it does not exist in closed form?

There is of course -s function equation and alternating series form but what about actual closed form? Valery Zapolodov (talk) 10:13, 4 January 2022 (UTC)

Okay, we know it does not from Hilbert. https://math.stackexchange.com/questions/1423720/is-there-any-proof-that-the-riemann-zeta-function-is-not-elementary Valery Zapolodov (talk) 10:25, 4 January 2022 (UTC)

Should we mention there is an essential singularity on Infinity?

On Riemann sphere that is? Valery Zapolodov (talk) 13:50, 4 January 2022 (UTC)

I would say an essential singularity is ordinarily defined as being a type of isolated singularity, and infinity is not an isolated singularity of ζ. Adumbrativus (talk) 01:14, 5 January 2022 (UTC)

But on Riemann Sphere it is a point or what is called Complex Infinity. And R. zeta function has an essential singularity there. See Zeros and poles. Neither ζ(1/z) nor 1/ζ(1/z) have limit on 0. Valery Zapolodov (talk) 10:07, 5 January 2022 (UTC)

Oops, sorry, please ignore my comment. Adumbrativus (talk) 10:42, 5 January 2022 (UTC)

An easy way to decide "should we mention this?" is to ask first "do reliable sources mention this?" and if so, to add it with a citation. --JBL (talk) 11:51, 5 January 2022 (UTC)

Added a cite with a quote. https://link.springer.com/article/10.1007/s40315-020-00316-x Valery Zapolodov (talk) 14:24, 5 January 2022 (UTC)

Recent result

@Jomu5221, Tanfer Tanriverdi, and JayBeeEll: I just removed a recently added result from the article because it is not notable:

• it is not a standard or well-known result
• the paper announcing it has a citation count of zero in Google Scholar, indicating that it is not notable (yet?)
• the paper was published in The Mathematical Gazette, which is a journal for teachers and learners of mathematics
• there's no hurry -- we can wait to see if this turns out to be an important result

The recent addition of a claim about "the first work that led to the derivation of the formula" is even less suitable for the article:

• it was inserted by its author (WP:COI)
• there is no third-party source confirming its importance

Cheers, --Macrakis (talk) 00:28, 17 January 2022 (UTC)

Thanks. Comparing name of the author of the Gazette article with the username of the account that added it, I think COI is a relevant concern there as well. I support your removal. --JBL (talk) 01:01, 17 January 2022 (UTC)

Log base template

I am fairly certain that this article uses log() to denote logarithms in base e rather than base 10. If so, please confirm this and I will add the following template: Toadspike (talk) 16:32, 12 May 2022 (UTC)

Possible Dirichlet series issue?

The first series under the "Representation" section does not seem to converge correctly (e.g. near zero, it converges to zero when it should converge to -1/2). Other representations seem to work fine; indeed the very next series converges to -1/2 as expected. Moreover, I chased the given reference, and did not see (either) series listed in the given location. It is possible, perhaps likely, that I have made an error. — Preceding unsigned comment added by Geometrian (talk • contribs) 03:26, 2 June 2022 (UTC)

Why don't we mention p-series here?

https://en.wikipedia.org/w/index.php?title=Riemann_zeta_function&oldid=prev&diff=1102657060

Oh I guess I should've linked to this

https://en.wikipedia.org/wiki/Convergence_tests#p-series_test

instead of that

https://en.wikipedia.org/wiki/P_series

but still?

Thewriter006 (talk) 07:27, 6 August 2022 (UTC)