Talk:Harmonic number

Riemann Hypothesis

I think we should put the Riemann Hypothesis section back... Look at it this way: someone studying the harmonic numbers using our encyclopedia would miss out on the connection to the Riemann Hypothesis. Scythe33 20:10, 7 August 2005 (UTC)

What's the connection? There are a variety of statements equivalent to RH, which involve all sorts of constants, sums, limits and formulas. I didn't see any direct connection at all; the statement was a bound on the divisor function. I'd rather see all of the various RH-eqivalent statements written up in RH article directly; preferably at greater length even. (I mean, everything is connected to RH if you dig deep enough; I don't think its right to footnote everything to say "this is connected" ... almost every article I've edited on WP is a topic that is one-off from RH...) linas 20:49, 8 August 2005 (UTC)

I think the RH is much deeper and more complex than the harmonic numbers, and that the section on the RH equivalent is more relevent on the RH page itself. EulerGamma 23:16, 4 September 2006 (UTC)

Anyway, I edited out this:

"Note that $n$ may be equal to $\infty$, provided $m>1$.

And if $m\leq 1$, while $n=\infty$, the harmonic series does not :converge and hence the harmonic number does not exist."

because infinity is not a real number, and thus is not a natural number. One must be very careful when saying "</math>n = \infty</math>" in any formal mathematical context. EulerGamma 23:16, 4 September 2006 (UTC)

Also, I see that it talks about a recurrence relation with Euler's Identity. This is called induction, and all the article shows is that if it is true for n, it is true for n+1. It fails to mention that it is true for n = 1 (even though this is quite obvious). I have an idea for how to show the identity (does anybody think it is okay for inclusion?):

$H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}$

$=\sum _{k=1}^{n}\int _{0}^{1}x^{k-1}dx$

$=\int _{0}^{1}\sum _{k=1}^{n}x^{k-1}dx$

$=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}dx$

Another thing that I am thinking might be good for inclusion, is that (by the above method):

$\sum _{k=1}^{n}{\frac {a^{k}}{k}}=\int _{0}^{a}{\frac {1-x^{n}}{1-x}}dx$

LaTeX issue

I can't get the first math line of the section "Special values for fractional arguments" to be rendered properly like the other lines. I don't know why. Chymicus (talk) 23:20, 10 April 2008 (UTC)

How to compute specials values of fractional argument

If $n,m$ are natural numbers, then:

$H_{\frac {m}{n}}={\frac {n}{m}}-\ln \left(n\right)+H_{m-1}+\sum \limits _{p=1}^{n-1}{e^{\frac {i2\pi pm}{n}}\left(\ln \left(1-e^{-{\frac {i2\pi p}{n}}}\right)+\sum \limits _{k=1}^{m-1}{\frac {e^{-k{\frac {i2\pi p}{n}}}}{k}}\right)}$ .

It can be drived from the identity $H_{x}=\sum \limits _{k=1}^{\infty }{{\frac {1}{k}}-{\frac {1}{k+x}}}$

--77.125.151.16 (talk) 14:13, 3 January 2010 (UTC)

Error

The following statement does not make sense:

for every x > 0, integer or not. We have:

$\int _{0}^{1}H_{x}\,dx=\gamma ,$

where γ is the Euler–Mascheroni constant

Since the harmonic numbers are monotonically growing this cannot be true. I don't know what the correct one is, but this is pretty certainly wrong. — Preceding unsigned comment added by 129.78.233.211 (talk • contribs)

I think you may have misunderstood what is meant here. The phrase "for every x > 0, integer or not" belongs to the infinite series definition of the continuous harmonic function on the previous line. The next line then says that the integral of this function between 0 and 1 is γ. I have modified the article to make this clearer. Gandalf61 (talk) 08:08, 20 May 2013 (UTC)

Um

Does anyone notice that $H_{n}={\frac {1}{\frac {n\cdot (n+1)}{2}}}$, Maybe that can be added. 190.60.93.218 (talk) 13:34, 1 August 2013 (UTC)

Not sure why you think that, but your formula fails for n = 2, when $H_{2}=1+{\frac {1}{2}}={\frac {3}{2}}$. Actually, you formula obviously fails because it is less than 1 for n > 1, whereas $H_{n}$ is greater than 1 for n > 1. Gandalf61 (talk) 13:41, 1 August 2013 (UTC)

I have made a terrible mistake, however I'm glad you've corrected me, after all I'm just learning. 190.60.93.218 (talk) 15:02, 1 August 2013 (UTC)

Sum of Harmonic series as an infinite series - original research?

This section concerning Aakash Praliya's derivation appears to be original research due to its wording and since the image of the derivation (see below) was uploaded by the user AakashPraliya2. Note that almost the same wording also appears on teh article Harmonic_series_(mathematics)

— Preceding unsigned comment added by Falcor84 (talk • contribs) 19:28, 12 October 2013 (UTC)

Repairing recursions

Please, could someone more dignified than I am take care about adding either missing or more reasonable base cases in the recurrence relations in this article? Both my efforts to either generally have as the base case some $H_{0}=0,$ not particularly coined as one of the harmonic numbers, or to specifically introduce it at places in specific need, were promptly reverted, ignoring that $H_{0}=0$ is already in use at the end of this section. In particular, I suggest to have $H_{0}=0,$ and to restrict n to the positive integers $(n\geq 1),$ throughout the whole article. Then it is about

• this section, in the recurrence by definition, and for $f'_{n},$ both recurring to $n-1,$ and again in
• this section, where $\psi (n)$ involves $H_{n-1}.$

Some of the identities and formulae would remain true for $n=0,$ but I think keeping the $H_{0}$ outside the range of $H_{n}$ is more appropriate. Thanks for taking in consideration. Purgy (talk) 10:31, 10 January 2019 (UTC)

As for me, it is standard enough that $\textstyle \sum _{k=1}^{0}a_{k}=0$ irrespective of $a_{k},$ and in particular for $a_{k}={\frac {1}{k}}.$ If it is not standard to say $H_{0}=0,$ then we should define $\textstyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}$ for integer $n>0$ (rather than for all possible $n$). Boris Tsirelson (talk) 19:04, 10 January 2019 (UTC)

I think it is also worth observing that Euler's integral representation gives $H_{0}=0$, which is some sort of evidence that it's okay to use $H_{0}$ as a base case for recursions for $H_{n}$. Ozob (talk) 02:18, 11 January 2019 (UTC)

Thanks for all the expenditure that I triggered. May I, please, remark that I have no troubles with the value of an empty sum, that I tried to introduce $H_{0}=0,$ independent of $H_{n},$ that I advocated $n>0$ as valid throughout the whole article (not only for occurrences of $\log n$, but also as safety margin to ${\tfrac {1}{n}}$ in the definition with ellipsis, and disregarding the validity of most(?) other identities for $n=0$), but that my main concern is the naive desire to have at the specified places a valid base case in every recurrence, especially in those referring to $n-1,$ and the wish to allow for $\psi (1)=-\gamma ,$ which would involve an $H_{0}.$ Cordial thanks to all, nevertheless. Purgy (talk) 10:34, 11 January 2019 (UTC)

Well, now the other side is welcome to speak out... Boris Tsirelson (talk) 11:42, 11 January 2019 (UTC)

Any reasonable definition of the harmonic numbers should be expected to extend correctly to 0. The question is whether anyone has ever done such a thing in a reliable secondary source. If not, then we shouldn't either.

Also I have to admit that I find the concern about this rather odd. Evidently Purgy doesn't ask that every time the symbol H_n appear, we announce that n must be an integer; why may that be implicit but the fact that n is positive may not be? It is an extremely widespread, common, and sensible shorthand that an equality between two things with different domains is "whenever both make sense" -- this is particularly straightforward for the case of the digamma function. --JBL (talk) 13:18, 11 January 2019 (UTC)

It is indeed reasonably clear from the context that n is an integer, and nonnegative; the only problem is, including zero or excluding zero. As far as I understand, Purgy wants to dispel any doubt at this point, either (a) once and for all the article, or (b) case-by-case; and Joel did already (at 13:19) a step toward the approach (a). I mean, the domain of n may be made the same throughout the article by shifting n when needed. Boris Tsirelson (talk) 17:59, 11 January 2019 (UTC)

Stirling numbers, combinations

The article claims that the harmonic numbers are related to the Stirling numbers as:

$H_{n}={\frac {1}{n!}}\left[{n+1 \atop 2}\right].$

The first few terms of $\left[{n+1 \atop 2}\right]$ are

1, 3, 6, 10, 15, 21, 28, 36, 45

and the first few terms of $n!$ are

1, 2, 6, 24, 120, 720, 5040, 40320, 362880

The product of the two tends quickly toward zero, which is not what $H_{n}$ does. Maybe the square brackets are being used to indicate something other than "combinations?" --LesPaul75_talk 22:50, 9 March 2019 (UTC)

In the table at Stirling numbers of the first kind, the first few values of $\textstyle \left[{n+1 \atop 2}\right]$ are 1, 3, 11, 50, 274, ..., which agrees with the statement in the article. –Deacon Vorbis (carbon • videos) 23:07, 9 March 2019 (UTC)

Yes; see for instance here: $\textstyle \left[{n \atop k}\right]$ is the number of permutations of n elements with exactly k cycles, called the (unsigned) Stirling number of the first kind. Boris Tsirelson (talk) 23:11, 9 March 2019 (UTC)

Extension to the complex numbers: recent edits, alternate formulations, OR, etc.

The beginning of the following discussion is copied from my talk page, lightly edited. --JBL (talk) 18:43, 3 June 2019 (UTC)

Regarding your edit comment "I don't think it is true" – if you have the time and could come up with a counterexample that disproves my language, that would be great! From my perspective, I made the change without adding a citation because I thought it was obviously true. I guess that means my task is to persuade you that it is obvious or to dig up a citation. 165.225.38.131 (talk) 16:22, 31 May 2019 (UTC)

Hi 165.225.38.131,

As I understand it, you assert that the third bullet point $\lim _{m\rightarrow \infty }\left[H_{m}-H_{m+x}\right]=0$ implies $H_{0}=0$; is that correct? If so, I do not see how the one thing implies the other. In particular, it seems to me that for any function H that satisfies the second and third bullet points, the function $J(x):=17+H(x)$ does as well. If I have misunderstood you, I invite you to clarify.

All the best, JBL (talk) 16:29, 31 May 2019 (UTC)

Thank you for your quick response. In my edit I had changed the third bullet to

lim_m→+∞ (H_m+x − ∑ m
k=1 1/k) = 0 for all complex values x

Thus, the trick of adding 17 no longer applies. Thanks! 165.225.38.131 (talk) 16:37, 31 May 2019 (UTC)

Ok, I get it. Here is my analysis of the situation:

I agree that your version is equivalent to the previous version; in particular, that it is correct. (Thank you for pointing out my error.) Your version comes without a citation, but that is also true of the original version, so that's a wash. The advantage of your version over the original version is that there is one fewer condition. There are at least two disadvantages of your version over the original version: the variable m in the limit must be restricted to integers in your version, which is artificial; and the statement with the "usual" harmonic numbers is considerably less natural-seeming than the version that makes reference only to the function H.

Overall, I probably prefer the original version to yours on aesthetic grounds, but don't feel strongly about it. What really would be good is (1) a proper reference and (2) trimming the OR enthusiasm on display down to something more reasonable. Your change doesn't help with this, but it doesn't hurt, either. Anyhow, I leave it to you to decide what to do next.

Also, if you agree, I'd like to copy this over to the article talk page, since that's where other editors of that article might be expected to find it. All the best, JBL (talk) 02:00, 1 June 2019 (UTC)

Yes, please feel free to copy or move this discussion. Yes, I agree that some citations would make this much better. Mathematically, I find it nice when the necessary assumptions are as week as possible, and

lim_m→+∞ (H_m+x − ∑ m
k=1 1/k) = 0

is weaker than

lim_m→+∞ (H_m+x − H_m) = 0.

On the other hand, once the harmonic number function is extended then it turns out that the latter is true. That is, the latter is a stronger result. At some point I may take a stab at achieving the high points and avoiding the low points. 165.225.38.131 (talk) 18:07, 3 June 2019 (UTC)

Thanks. I'm not sure I agree about strength, but as I say I leave it to you -- if you reinstate the change, I won't object. Now that I've copied this discussion over, perhaps other editors will weigh in (about this specific question, or about the article section more generally). It might also be worth a look through the article's history to see if the person/people who originally added the section are still active and might give some insight on whether source exist or whether it was just pure OR. --JBL (talk) 18:43, 3 June 2019 (UTC)

Harmonic numbers have been around for centuries and I strongly suspect that anything about them that is simple enough to be in Wikipedia has already been discovered and published. But an existence proof for the citation is not the same as an actual citation! Alternatively, we could argue that the content is obvious to anyone with a Ph.D. in mathematical analysis and thus satisfies the routine calculation exception to WP:OR, but I am not so sure that that is in the spirit of that rule. Thus, if someone wants to remove the "Alternative, asymptotic formulation" section, it would be hard for me to object. Personally, I like the section despite these failings and I won't be the person who instigates that. 165.225.38.131 (talk) 19:48, 3 June 2019 (UTC)

I have gone ahead and made the edit. Please revise (or revert) as you see fit. Cheers! 165.225.38.131 (talk) 13:22, 5 June 2019 (UTC)

The statement that $x(\log(x)-1)$ is an integral of $\log(x)$ does not need a citation.

The statement that $x(\log(x)-1)$ is an integral of $\log(x)$ is a routine calculation for a first semester calculus student and should not need a citation.Johnwaylandbales (talk) 16:07, 16 September 2019 (UTC)