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 ) {\displaystyle \zeta (s)} and is named after the mathematician Bernhard Riemann. When the argument s {\displaystyle s} is a real number greater than one, the zeta function satisfies the equation
ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\,.}
It can therefore provide the sum of various convergent infinite series, such as ζ ( 2 ) = 1 1 2 + {\textstyle \zeta (2)={\frac {1}{1^{2}}}+}
1 2 2 + {\textstyle {\frac {1}{2^{2}}}+}
1 3 2 + … . {\textstyle {\frac {1}{3^{2}}}+\ldots \,.} Explicit or numerically efficient formulae exist for ζ ( s ) {\displaystyle \zeta (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 {\displaystyle s} above also holds when s {\displaystyle 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 {\displaystyle 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 {\displaystyle s} , for which the corresponding summation would diverge. For example, the full zeta function exists at s = − 1 {\displaystyle s=-1} (and is therefore finite there), but the corresponding series would be 1 + 2 + 3 + … {\displaystyle 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 {\displaystyle s=-2,-4} , etc.), for which ζ ( s ) = 0 {\displaystyle \zeta (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
ζ ( 0 ) = B 1 − = − B 1 + = − 1 2 {\displaystyle \zeta (0)={B_{1}^{-}}=-{B_{1}^{+}}=-{\tfrac {1}{2}}\!}
At 1 there is a pole, so ζ ( 1 ) {\displaystyle \zeta (1)} is not finite but the left and right limits are:
lim ε → 0 ± ζ ( 1 + ε ) = ± ∞ {\displaystyle \lim _{\varepsilon \to 0^{\pm }}\zeta (1+\varepsilon )=\pm \infty }
Since it is a pole of first order, it has a complex residue
lim ε → 0 ε ζ ( 1 + ε ) = 1 . {\displaystyle \lim _{\varepsilon \to 0}\varepsilon \zeta (1+\varepsilon )=1\,.}
Positive integers
Even positive integers For the even positive integers n {\displaystyle n} , one has the relationship to the Bernoulli numbers B n {\displaystyle B_{n}} :
ζ ( n ) = ( − 1 ) n 2 + 1 ( 2 π ) n B n 2 ( n ! ) . {\displaystyle \zeta (n)=(-1)^{{\tfrac {n}{2}}+1}{\frac {(2\pi )^{n}B_{n}}{2(n!)}}\,.}
The computation of ζ ( 2 ) {\displaystyle \zeta (2)} is known as the Basel problem. The value of ζ ( 4 ) {\displaystyle \zeta (4)} is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:
ζ ( 2 ) = 1 + 1 2 2 + 1 3 2 + ⋯ = π 2 6 ζ ( 4 ) = 1 + 1 2 4 + 1 3 4 + ⋯ = π 4 90 ζ ( 6 ) = 1 + 1 2 6 + 1 3 6 + ⋯ = π 6 945 ζ ( 8 ) = 1 + 1 2 8 + 1 3 8 + ⋯ = π 8 9450 ζ ( 10 ) = 1 + 1 2 10 + 1 3 10 + ⋯ = π 10 93555 ζ ( 12 ) = 1 + 1 2 12 + 1 3 12 + ⋯ = 691 π 12 638512875 ζ ( 14 ) = 1 + 1 2 14 + 1 3 14 + ⋯ = 2 π 14 18243225 ζ ( 16 ) = 1 + 1 2 16 + 1 3 16 + ⋯ = 3617 π 16 325641566250 . {\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}}\\[4pt]\zeta (16)&=1+{\frac {1}{2^{16}}}+{\frac {1}{3^{16}}}+\cdots ={\frac {3617\pi ^{16}}{325641566250}}\,.\end{aligned}}}
Taking the limit n → ∞ {\displaystyle n\rightarrow \infty } , one obtains ζ ( ∞ ) = 1 {\displaystyle \zeta (\infty )=1} .
The relationship between zeta at the positive even integers and powers of pi may be written as
a n ζ ( 2 n ) = π 2 n b n {\displaystyle a_{n}\zeta (2n)=\pi ^{2n}b_{n}}
where a n {\displaystyle a_{n}} and b n {\displaystyle b_{n}} are coprime positive integers for all n {\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 η n = b n / a n {\displaystyle \eta _{n}=b_{n}/a_{n}} be the coefficient of π 2 n {\displaystyle \pi ^{2n}} as above,
ζ ( 2 n ) = ∑ ℓ = 1 ∞ 1 ℓ 2 n = η n π 2 n {\displaystyle \zeta (2n)=\sum _{\ell =1}^{\infty }{\frac {1}{\ell ^{2n}}}=\eta _{n}\pi ^{2n}}
then we find recursively,
η 1 = 1 / 6 η n = ∑ ℓ = 1 n − 1 ( − 1 ) ℓ − 1 η n − ℓ ( 2 ℓ + 1 ) ! + ( − 1 ) n + 1 n ( 2 n + 1 ) ! {\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:
ζ ( 2 n ) = 1 n + 1 2 ∑ k = 1 n − 1 ζ ( 2 k ) ζ ( 2 n − 2 k ) for n > 1 {\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 d d x cot ( x ) = − 1 − cot 2 ( x ) {\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:
∑ n = 0 ∞ ζ ( 2 n ) x 2 n = − π x 2 cot ( π x ) = − 1 2 + π 2 6 x 2 + π 4 90 x 4 + π 6 945 x 6 + ⋯ {\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
lim n → ∞ ζ ( 2 n ) = 1 {\displaystyle \lim _{n\rightarrow \infty }\zeta (2n)=1} The formula also shows that for n ∈ N , n → ∞ {\displaystyle n\in \mathbb {N} ,n\rightarrow \infty } ,
| B 2 n | ∼ ( 2 n ) ! 2 ( 2 π ) 2 n {\displaystyle \left|B_{2n}\right|\sim {\frac {(2n)!\,2}{\;~(2\pi )^{2n}\,}}}
Odd positive integers The sum of the harmonic series is infinite.
ζ ( 1 ) = 1 + 1 2 + 1 3 + ⋯ = ∞ {\displaystyle \zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty \!}
The value ζ ( 3 ) {\displaystyle \zeta (3)} is also known as Apéry's constant and has a role in the electron's gyromagnetic ratio. The value ζ ( 3 ) {\displaystyle \zeta (3)} also appears in Planck's law. These and additional values are:
It is known that ζ ( 3 ) {\displaystyle \zeta (3)} is irrational (Apéry's theorem) and that infinitely many of the numbers ζ ( 2 n + 1 ) : n ∈ N {\displaystyle \zeta (2n+1):n\in \mathbb {N} } , 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 ζ ( 5 ) , ζ ( 7 ) , ζ ( 9 ) , {\displaystyle \zeta (5),\zeta (7),\zeta (9),} or ζ ( 11 ) {\displaystyle \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)
ζ ( 5 ) = 1 294 π 5 − 72 35 ∑ n = 1 ∞ 1 n 5 ( e 2 π n − 1 ) − 2 35 ∑ n = 1 ∞ 1 n 5 ( e 2 π n + 1 ) ζ ( 5 ) = 12 ∑ n = 1 ∞ 1 n 5 sinh ( π n ) − 39 20 ∑ n = 1 ∞ 1 n 5 ( e 2 π n − 1 ) + 1 20 ∑ n = 1 ∞ 1 n 5 ( e 2 π n + 1 ) {\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)
ζ ( 7 ) = 19 56700 π 7 − 2 ∑ n = 1 ∞ 1 n 7 ( e 2 π n − 1 ) {\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
S ± ( s ) = ∑ n = 1 ∞ 1 n s ( e 2 π n ± 1 ) {\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
0 = a n ζ ( n ) − b n π n + c n S − ( n ) + d n S + ( n ) {\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 {\displaystyle a_{n},b_{n},c_{n}} and d n {\displaystyle 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 In general, for negative integers, one has
ζ ( − n ) = − B n + 1 n + 1 {\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}
The so-called "trivial zeros" occur at the negative even integers:
ζ ( − 2 n ) = 0 {\displaystyle \zeta (-2n)=0} (Ramanujan summation) The first few values for negative odd integers are
ζ ( − 1 ) = − 1 12 ζ ( − 3 ) = 1 120 ζ ( − 5 ) = − 1 252 ζ ( − 7 ) = 1 240 ζ ( − 9 ) = − 1 132 ζ ( − 11 ) = 691 32760 ζ ( − 13 ) = − 1 12 {\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}}}
This means ζ ( m ) {\displaystyle \zeta (m)} can be used as the definition of Bernoulli numbers. However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on ζ ( − 1 ) {\displaystyle \zeta (-1)} , see 1 + 2 + 3 + 4