Particular values of the Riemann zeta function

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 first few values are given by:

${\displaystyle \zeta (2)=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}=1.6449\dots \!}$ (OEIS: A013661)

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

${\displaystyle \zeta (4)=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}=1.0823\dots \!}$ (OEIS: A013662)

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

${\displaystyle \zeta (6)=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}=1.0173\dots \!}$ (OEIS: A013664)

${\displaystyle \zeta (8)=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}=1.00407\dots \!}$ (OEIS: A013666)

${\displaystyle \zeta (10)=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}=1.000994\dots \!}$ (OEIS: A013668)

${\displaystyle \zeta (12)=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}=1.000246\dots \!}$ (OEIS: A013670)

${\displaystyle \zeta (14)=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}=1.0000612\dots \!}$ (OEIS: A013672).

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

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 OEIS: A002432 and OEIS: A046988, respectively, in OEIS. Some of these values are reproduced below:

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 ${\displaystyle n\in \mathbb {N} ,n\rightarrow \infty }$,

${\displaystyle \left|B_{2n}\right|\sim {\frac {(2n)!\,2}{\;~(2\pi )^{2n}\,}}}$

For the first few odd natural numbers one has

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

(the harmonic series);

${\displaystyle \zeta (3)=1+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+\cdots =1.20205\dots \!}$ (OEIS: A02117)

(Called Apéry's constant and has a role in the electron's gyromagnetic ratio)

${\displaystyle \zeta (5)=1+{\frac {1}{2^{5}}}+{\frac {1}{3^{5}}}+\cdots =1.03692\dots \!}$ (OEIS: A013663)

(Appears in Planck's law)

${\displaystyle \zeta (7)=1+{\frac {1}{2^{7}}}+{\frac {1}{3^{7}}}+\cdots =1.00834\dots \!}$ (OEIS: A013665)

${\displaystyle \zeta (9)=1+{\frac {1}{2^{9}}}+{\frac {1}{3^{9}}}+\cdots =1.002008\dots \!}$ (OEIS: 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.

Plouffe gives the following identities

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

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

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

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 A_n, B_n, C_n and D_n are positive integers. Plouffe gives a table of values:

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]