Particular values of the Riemann zeta function

This article gives some specific values of the Riemann zeta function, including values at integer arguments, and some series involving them.

The Riemann zeta function at 0 and 1

At zero, one has

\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:

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

Since it is a pole of first order, its principal value exists and is equal to the Euler–Mascheroni constant γ = 0.57721 56649+.

Positive integers

Even positive integers

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

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

for n ∈ N. The first few values are given by:

\zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}=1.6449\dots \!

(

A013661)

(the demonstration of this equality is known as the Basel problem)

\zeta (4)=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}=1.0823\dots \!

(

A013662)

(the Stefan–Boltzmann law and Wien approximation in physics)

\zeta (6)=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}=1.0173...\dots \!

(

A013664)

\zeta (8)=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}=1.00407...\dots \!

(

A013666)

\zeta (10)=1+{\frac {1}{2^{{10}}}}+{\frac {1}{3^{{10}}}}+\cdots ={\frac {\pi ^{{10}}}{93555}}=1.000994...\dots \!

(

A013668)

\zeta (12)=1+{\frac {1}{2^{{12}}}}+{\frac {1}{3^{{12}}}}+\cdots ={\frac {691\pi ^{{12}}}{638512875}}=1.000246\dots \!

(

A013670)

\zeta (14)=1+{\frac {1}{2^{{14}}}}+{\frac {1}{3^{{14}}}}+\cdots ={\frac {2\pi ^{{14}}}{18243225}}=1.0000612\dots \!

(

A013672).

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

A_{n}\zeta (n)=B_{n}\pi ^{n}\,\!

where A_n and B_n are integers for all even n. These are given by the integer sequences A002432 and A046988, respectively, in OEIS. Some of these values are reproduced below:

coefficients
n	A	B
2	6	1
4	90	1
6	945	1
8	9450	1
10	93555	1
12	638512875	691
14	18243225	2
16	325641566250	3617
18	38979295480125	43867
20	1531329465290625	174611
22	13447856940643125	155366
24	201919571963756521875	236364091
26	11094481976030578125	1315862
28	564653660170076273671875	6785560294
30	5660878804669082674070015625	6892673020804
32	62490220571022341207266406250	7709321041217
34	12130454581433748587292890625	151628697551

If we let η_n be the coefficient B/A as above,

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

then we find recursively,

{\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:

\zeta (2n)={\frac {1}{n+{\frac {1}{2}}}}\sum _{{k=1}}^{{n-1}}\zeta (2k)\zeta (2n-2k),n>1

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

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

\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

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

The formula also shows that for $n\in {\mathbb {N}},n\rightarrow \infty$ ,

\left|B_{{2n}}\right|\sim {\frac {2(2n)!}{(2\pi )^{{2n}}}}

Odd positive integers

For the first few odd natural numbers one has

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

(the harmonic series);

\zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.20205\dots \!

(Apéry's constant)

\zeta (5)=1+{\frac {1}{2^{5}}}+{\frac {1}{3^{5}}}+\cdots =1.03692\dots \!

(

A013663)

\zeta (7)=1+{\frac {1}{2^{7}}}+{\frac {1}{3^{7}}}+\cdots =1.00834\dots \!

(

A013665)

\zeta (9)=1+{\frac {1}{2^{9}}}+{\frac {1}{3^{9}}}+\cdots =1.002008\dots \!

(

A013667)

It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) (n ∈ 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

{\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)

\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

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

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

where A_n, B_n, C_n and D_n 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 including zero, one has

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

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

\zeta (-2n)=0\,

The first few values for negative odd integers are

{\begin{aligned}\zeta (-1)=-{\frac {1}{12}}\\[6pt]\zeta (-3)={\frac {1}{120}}\\[6pt]\zeta (-5)=-{\frac {1}{252}}\\[6pt]\zeta (-7)={\frac {1}{240}}\\[6pt]\zeta (-9)=-{\frac {1}{132}}\\[6pt]\zeta (-11)={\frac {691}{32760}}\\[6pt]\zeta (-13)=-{\frac {1}{12}}\\[6pt]\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

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

The first few values of which are

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

One also has

\zeta ^{{\prime }}(0)=-{\frac {1}{2}}\ln(2\pi )\approx -0.918938533\ldots

(

A075700),

\zeta ^{{\prime }}(-1)={\frac {1}{12}}-\ln A\approx -0.1654211437\ldots

(

A084448)

and

\zeta ^{\prime }(2)={\frac {1}{6}}\pi ^{2}(\gamma +\ln 2-12\ln A+\ln \pi )\approx -0.93754825\ldots

(

A073002)

where A is the Glaisher–Kinkelin constant.

Series involving ζ(n)

The following sums can be derived from the generating function:

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

where ψ₀ is the digamma function.

\sum _{{k=2}}^{\infty }(\zeta (k)-1)=1

\sum _{{k=1}}^{\infty }(\zeta (2k)-1)={\frac {3}{4}}

\sum _{{k=1}}^{\infty }(\zeta (2k+1)-1)={\frac {1}{4}}

\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)={\frac {1}{2}}

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

\sum _{{k=2}}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}=\gamma

\sum _{{k=2}}^{\infty }{\frac {\zeta (k)-1}{k}}=1-\gamma

\sum _{{k=2}}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}=\ln 2+\gamma -1

and using the principal value

\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

\sum _{{k=1}}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}=0

\sum _{{k=1}}^{\infty }{\frac {\zeta (k)-1}{k}}=0

\sum _{{k=1}}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}=\ln 2

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.

References

↑ 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. Mathématique. 331: 267–270. arXiv:math/0008051 . doi:10.1016/S0764-4442(00)01624-4.
↑ W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. doi:10.1070/rm2001v056n04abeh000427.
↑ Boos, H. E.; Korepin, V. E.; Nishiyama, Y.; Shiroishi, M. (2002), "Quantum correlations and number theory", J. Phys. A, 35: 4443–4452, arXiv:cond-mat/0202346  .
↑ 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.
↑ E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).
↑ E. A. Karatsuba: Fast evaluation of ζ(3). Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).