Particular values of the Riemann zeta function

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In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ(s) and is named after the mathematician Bernhard Riemann. When the argument s is a real number greater than one, the zeta function satisfies the equation

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\,.}$
It can therefore provide the sum of various convergent infinite series, such as ${\textstyle \zeta (2)={\frac {1}{1^{2}}}+}$${\textstyle {\frac {1}{2^{2}}}+}$${\textstyle {\frac {1}{3^{2}}}+\ldots \,.}$ Explicit or numerically efficient formulae exist for ζ(s) at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.

The same equation in s above also holds when s is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at s = 1. The complex derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values of s, for which the corresponding summation would diverge. For example, the full zeta function exists at s = −1 (and is therefore finite there), but the corresponding series would be ${\textstyle 1+2+3+\ldots \,,}$ whose partial sums would grow indefinitely large.

The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.

The Riemann zeta function at 0 and 1

At zero, one has

${\displaystyle \zeta (0)={B_{1}^{-}}=-{B_{1}^{+}}=-{\tfrac {1}{2}}\!}$

At 1 there is a pole, so ζ(1) is not finite but the left and right limits are:

${\displaystyle \lim _{\varepsilon \to 0^{\pm }}\zeta (1+\varepsilon )=\pm \infty }$

Since it is a pole of first order, it has a complex residue
${\displaystyle \lim _{\varepsilon \to 0}\varepsilon \zeta (1+\varepsilon )=1\,.}$

Positive integers

Even positive integers

For the even positive integers, one has the relationship to the Bernoulli numbers:

${\displaystyle \zeta (2n)=(-1)^{n+1}{\frac {(2\pi )^{2n}B_{2n}}{2(2n)!}}\!}$

for ${\displaystyle n\in \mathbb {N} }$ .

The computation of ζ(2) is known as the Basel problem. The value of ζ(4) is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:

{\displaystyle {\begin{aligned}\zeta (2)&=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\\[4pt]\zeta (4)&=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\\[4pt]\zeta (6)&=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}\\[4pt]\zeta (8)&=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}\\[4pt]\zeta (10)&=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}\\[4pt]\zeta (12)&=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}\\[4pt]\zeta (14)&=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}\,.\end{aligned}}}

Taking the limit ${\displaystyle n\rightarrow \infty }$ , one obtains ${\displaystyle \zeta (\infty )=1}$ .

Selected values for even integers
Value Decimal expansion Source
${\displaystyle \zeta (2)}$  1.6449340668482264364...
${\displaystyle \zeta (4)}$  1.0823232337111381915...
${\displaystyle \zeta (6)}$  1.0173430619844491397...
${\displaystyle \zeta (8)}$  1.0040773561979443393...
${\displaystyle \zeta (10)}$  1.0009945751278180853...
${\displaystyle \zeta (12)}$  1.0002460865533080482...
${\displaystyle \zeta (14)}$  1.0000612481350587048...

The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as

${\displaystyle A_{n}\zeta (2n)=\pi ^{2n}B_{n}}$

where ${\displaystyle A_{n}}$  and ${\displaystyle B_{n}}$  are integers for all even ${\displaystyle n}$ . These are given by the integer sequences and , respectively, in OEIS. Some of these values are reproduced below:

coefficients
n A B
1 6 1
2 90 1
3 945 1
4 9450 1
5 93555 1
6 638512875 691
7 18243225 2
8 325641566250 3617
9 38979295480125 43867
10 1531329465290625 174611
11 13447856940643125 155366
12 201919571963756521875 236364091
13 11094481976030578125 1315862
14 564653660170076273671875 6785560294
15 5660878804669082674070015625 6892673020804
16 62490220571022341207266406250 7709321041217
17 12130454581433748587292890625 151628697551

If we let ${\displaystyle \eta _{n}=B_{n}/A_{n}}$  be the coefficient of ${\displaystyle \pi ^{2n}}$  as above,

${\displaystyle \zeta (2n)=\sum _{\ell =1}^{\infty }{\frac {1}{\ell ^{2n}}}=\eta _{n}\pi ^{2n}}$

then we find recursively,

{\displaystyle {\begin{aligned}\eta _{1}&=1/6\\\eta _{n}&=\sum _{\ell =1}^{n-1}(-1)^{\ell -1}{\frac {\eta _{n-\ell }}{(2\ell +1)!}}+(-1)^{n+1}{\frac {n}{(2n+1)!}}\end{aligned}}}

This recurrence relation may be derived from that for the Bernoulli numbers.

Also, there is another recurrence:

${\displaystyle \zeta (2n)={\frac {1}{n+{\frac {1}{2}}}}\sum _{k=1}^{n-1}\zeta (2k)\zeta (2n-2k)\quad {\text{ for }}\quad n>1}$

which can be proved, using that ${\displaystyle {\frac {d}{dx}}\cot(x)=-1-\cot ^{2}(x)}$

The values of the zeta function at non-negative even integers have the generating function:

${\displaystyle \sum _{n=0}^{\infty }\zeta (2n)x^{2n}=-{\frac {\pi x}{2}}\cot(\pi x)=-{\frac {1}{2}}+{\frac {\pi ^{2}}{6}}x^{2}+{\frac {\pi ^{4}}{90}}x^{4}+{\frac {\pi ^{6}}{945}}x^{6}+\cdots }$

Since
${\displaystyle \lim _{n\rightarrow \infty }\zeta (2n)=1}$

The formula also shows that for ${\displaystyle n\in \mathbb {N} ,n\rightarrow \infty }$ ,
${\displaystyle \left|B_{2n}\right|\sim {\frac {(2n)!\,2}{\;~(2\pi )^{2n}\,}}}$

Odd positive integers

The sum of the harmonic series is infinite.

${\displaystyle \zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty \!}$

The value ζ(3) is also known as Apéry's constant and has a role in the electron's gyromagnetic ratio. The value ζ(3) appears in Planck's law. These and additional values are:

Selected values for odd integers
Value Decimal expansion Source
${\displaystyle \zeta (3)}$  1.2020569031595942853...
${\displaystyle \zeta (5)}$  1.0369277551433699263...
${\displaystyle \zeta (7)}$  1.0083492773819228268...
${\displaystyle \zeta (9)}$  1.0020083928260822144...
${\displaystyle \zeta (11)}$  1.0004941886041194645...
${\displaystyle \zeta (13)}$  1.0001227133475784891...
${\displaystyle \zeta (15)}$  1.0000305882363070204...

It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n${\displaystyle \mathbb {N} }$  , are irrational.[1] There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ(5), ζ(7), ζ(9), or ζ(11) is irrational.[2]

The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.[3]

Most of the identities following below are provided by Simon Plouffe. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.

ζ(5)

Plouffe gives the following identities

{\displaystyle {\begin{aligned}\zeta (5)&={\frac {1}{294}}\pi ^{5}-{\frac {72}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {2}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\\\zeta (5)&=12\sum _{n=1}^{\infty }{\frac {1}{n^{5}\sinh(\pi n)}}-{\frac {39}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {1}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\end{aligned}}}

ζ(7)

${\displaystyle \zeta (7)={\frac {19}{56700}}\pi ^{7}-2\sum _{n=1}^{\infty }{\frac {1}{n^{7}(e^{2\pi n}-1)}}\!}$

Note that the sum is in the form of a Lambert series.

ζ(2n + 1)

By defining the quantities

${\displaystyle S_{\pm }(s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}(e^{2\pi n}\pm 1)}}}$

a series of relationships can be given in the form

${\displaystyle 0=A_{n}\zeta (n)-B_{n}\pi ^{n}+C_{n}S_{-}(n)+D_{n}S_{+}(n)}$

where An, Bn, Cn and Dn are positive integers. Plouffe gives a table of values:

coefficients
n A B C D
3 180 7 360 0
5 1470 5 3024 84
7 56700 19 113400 0
9 18523890 625 37122624 74844
11 425675250 1453 851350500 0
13 257432175 89 514926720 62370
15 390769879500 13687 781539759000 0
17 1904417007743250 6758333 3808863131673600 29116187100
19 21438612514068750 7708537 42877225028137500 0
21 1881063815762259253125 68529640373 3762129424572110592000 1793047592085750

These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.

A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.[4][5][6]

Negative integers

In general, for negative integers (and also zero), one has

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

The so-called "trivial zeros" occur at the negative even integers:

${\displaystyle \zeta (-2n)=0}$

(Ramanujan summation)

The first few values for negative odd integers are

{\displaystyle {\begin{aligned}\zeta (-1)&=-{\frac {1}{12}}\\[4pt]\zeta (-3)&={\frac {1}{120}}\\[4pt]\zeta (-5)&=-{\frac {1}{252}}\\[4pt]\zeta (-7)&={\frac {1}{240}}\\[4pt]\zeta (-9)&=-{\frac {1}{132}}\\[4pt]\zeta (-11)&={\frac {691}{32760}}\\[4pt]\zeta (-13)&=-{\frac {1}{12}}\end{aligned}}}

However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on the first value, see 1 + 2 + 3 + 4 + · · ·.

So ζ(m) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.

Derivatives

The derivative of the zeta function at the negative even integers is given by

${\displaystyle \zeta ^{\prime }(-2n)=(-1)^{n}{\frac {(2n)!}{2(2\pi )^{2n}}}\zeta (2n+1)\,.}$

The first few values of which are

{\displaystyle {\begin{aligned}\zeta ^{\prime }(-2)&=-{\frac {\zeta (3)}{4\pi ^{2}}}\\[4pt]\zeta ^{\prime }(-4)&={\frac {3}{4\pi ^{4}}}\zeta (5)\\[4pt]\zeta ^{\prime }(-6)&=-{\frac {45}{8\pi ^{6}}}\zeta (7)\\[4pt]\zeta ^{\prime }(-8)&={\frac {315}{4\pi ^{8}}}\zeta (9)\,.\end{aligned}}}

One also has

{\displaystyle {\begin{aligned}\zeta ^{\prime }(0)&=-{\frac {1}{2}}\ln(2\pi )\\[4pt]\zeta ^{\prime }(-1)&={\frac {1}{12}}-\ln A\\[4pt]\zeta ^{\prime }(2)&={\frac {1}{6}}\pi ^{2}(\gamma +\ln 2-12\ln A+\ln \pi )\end{aligned}}}

where A is the Glaisher–Kinkelin constant.

From the logarithmic derivative of the functional equation,

${\displaystyle 2{\frac {\zeta '(1/2)}{\zeta (1/2)}}=\log(2\pi )+{\frac {\pi \cos(\pi /4)}{2\sin(\pi /4)}}-{\frac {\Gamma '(1/2)}{\Gamma (1/2)}}=\log(2\pi )+{\frac {\pi }{2}}+2\log 2+\gamma \,.}$

Selected derivatives
Value Decimal expansion Source
${\displaystyle \zeta '(3)}$  −0.19812624288563685333...
${\displaystyle \zeta '(2)}$  −0.93754825431584375370...
${\displaystyle \zeta '(0)}$  −0.91893853320467274178...
${\displaystyle \zeta '(-{\tfrac {1}{2}})}$  −0.36085433959994760734...
${\displaystyle \zeta '(-1)}$  −0.16542114370045092921...
${\displaystyle \zeta '(-2)}$  −0.030448457058393270780...
${\displaystyle \zeta '(-3)}$  +0.0053785763577743011444...
${\displaystyle \zeta '(-4)}$  +0.0079838114502686242806...
${\displaystyle \zeta '(-5)}$  −0.00057298598019863520499...
${\displaystyle \zeta '(-6)}$  −0.0058997591435159374506...
${\displaystyle \zeta '(-7)}$  −0.00072864268015924065246...
${\displaystyle \zeta '(-8)}$  +0.0083161619856022473595...

Series involving ζ(n)

The following sums can be derived from the generating function:

${\displaystyle \sum _{k=2}^{\infty }\zeta (k)x^{k-1}=-\psi _{0}(1-x)-\gamma }$

where ψ0 is the digamma function.

{\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(\zeta (k)-1)&=1\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k)-1)&={\frac {3}{4}}\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k+1)-1)&={\frac {1}{4}}\\[4pt]\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)&={\frac {1}{2}}\end{aligned}}}

Series related to the Euler–Mascheroni constant (denoted by γ) are

{\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=\gamma \\[4pt]\sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}&=1-\gamma \\[4pt]\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2+\gamma -1\end{aligned}}}

and using the principal value

${\displaystyle \zeta (k)=\lim _{\varepsilon \to 0}{\frac {\zeta (k+\varepsilon )+\zeta (k-\varepsilon )}{2}}}$

which of course affects only the value at 1, these formulae can be stated as

{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }{\frac {\zeta (k)-1}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2\end{aligned}}}

and show that they depend on the principal value of ζ(1) = γ .

Nontrivial zeros

Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". See Andrew Odlyzko's website for their tables and bibliographies.

Ratios

Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting particular values of the gamma function into the functional equation

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

We have simple relations for half-integer arguments

{\displaystyle {\begin{aligned}{\frac {\zeta (3/2)}{\zeta (-1/2)}}&=-4\pi \\{\frac {\zeta (5/2)}{\zeta (-3/2)}}&=-{\frac {16\pi ^{2}}{3}}\\{\frac {\zeta (7/2)}{\zeta (-5/2)}}&={\frac {64\pi ^{3}}{15}}\\{\frac {\zeta (9/2)}{\zeta (-7/2)}}&={\frac {256\pi ^{4}}{105}}\end{aligned}}}

Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation

${\displaystyle \Gamma \left({\tfrac {3}{4}}\right)=\left({\tfrac {\pi }{2}}\right)^{\tfrac {1}{4}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}^{\tfrac {1}{2}}}$

is the zeta ratio relation

${\displaystyle {\frac {\zeta (3/4)}{\zeta (1/4)}}=2{\sqrt {\frac {\pi }{(2-{\sqrt {2}})\operatorname {AGM} \left({\sqrt {2}},1\right)}}}}$

where AGM is the arithmetic–geometric mean. In a similar vein, it is possible to form radical relations, such as from

${\displaystyle {\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}}$

the analogous zeta relation is

${\displaystyle {\frac {\zeta (1/5)^{2}\zeta (7/10)\zeta (9/10)}{\zeta (1/10)\zeta (3/10)\zeta (4/5)^{2}}}={\frac {(5-{\sqrt {5}})\left({\sqrt {10}}+{\sqrt {5+{\sqrt {5}}}}\right)}{10\cdot 2^{\tfrac {3}{10}}}}}$

References

1. ^ Rivoal, T. (2000). "La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs". Comptes Rendus de l'Académie des Sciences, Série I. 331: 267–270. arXiv:math/0008051. Bibcode:2000CRASM.331..267R. doi:10.1016/S0764-4442(00)01624-4. S2CID 119678120.
2. ^ W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/rm2001v056n04abeh000427.
3. ^ Boos, H.E.; Korepin, V.E.; Nishiyama, Y.; Shiroishi, M. (2002). "Quantum correlations and number theory". J. Phys. A. 35 (20): 4443–4452. arXiv:cond-mat/0202346. Bibcode:2002JPhA...35.4443B. doi:10.1088/0305-4470/35/20/305. S2CID 119143600..
4. ^ Karatsuba, E. A. (1995). "Fast calculation of the Riemann zeta function ζ(s) for integer values of the argument s". Probl. Perdachi Inf. 31 (4): 69–80. MR 1367927.
5. ^ E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).
6. ^ E. A. Karatsuba: Fast evaluation of ζ(3). Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).