Even positive integers For the even positive integers , one has the relationship to the
Bernoulli numbers : \zeta(n) = (-1)^{\tfrac{n}{2}+1}\frac{(2\pi)^{n}B_n}{2(n!)} \,. The computation of \zeta(2) is known as the
Basel problem. The value of \zeta(4) is related to the
Stefan–Boltzmann law and
Wien approximation in physics. The first few values are given by: \begin{align} \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} \\[4pt] \zeta(16) & = 1 + \frac{1}{2^{16}} + \frac{1}{3^{16}} + \cdots = \frac{3617\pi^{16}}{325641566250}\,. \end{align} Taking the limit , one obtains . The relationship between zeta at the positive even integers and powers of pi may be written as a_n \zeta(2n) = \pi^{2n} b_n where a_n and b_n are coprime positive integers for all n. These are given by the integer sequences and , respectively, in
OEIS. Some of these values are reproduced below: If we let \eta_n=b_n/a_n be the coefficient of \pi^{2n} as above, \zeta(2n) = \sum_{\ell=1}^{\infty}\frac{1}{\ell^{2n}}=\eta_n\pi^{2n} then we find recursively, \begin{align} \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{align} 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) \quad \text{ for } \quad 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{(2n)!\,2}{\;~(2\pi)^{2n}\,}
Odd positive integers The sum of the
harmonic series is infinite. \zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty\! The value \zeta(3) is also known as
Apéry's constant and has a role in the electron's
gyromagnetic ratio. The value \zeta(3) also appears in
Planck's law. These and additional values are: It is known that \zeta(3) is irrational (
Apéry's theorem) and that infinitely many of the numbers , are irrational. 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 \zeta(5), \zeta(7), \zeta(9), or \zeta(11) is irrational. The positive odd integers of the zeta function appear in physics, specifically
correlation functions of antiferromagnetic
XXX spin chain. 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 stated the following identities without proof. Proofs were later given by other authors.
ζ(5) \begin{align} \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{align}
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: 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. == Negative integers ==