Numbers, constants and computation

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The Riemann Zeta-function ζ(s) : generalities Xavier Gourdon and Pascal Sebah August 19, 20041

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Definition

The Zeta function was first introduced by Euler and is defined by ζ(s) =

∞ X 1 . ns n=1

(1)

The series is convergent when s is a complex number with 1. Some special values of ζ(s) are well known, for example the values ζ(2) = π 2 /6, ζ(4) = π 4 /90, were obtained by Euler. In 1859, Riemann had the idea to define ζ(s) for all complex number s by analytic continuation. This continuation is very important in number theory and plays a central role in the study of the distribution of prime numbers. Several techniques permit to extend the domain of definition of the Zeta function (the continuation is independant of the technique used because of uniqueness of analytic continuation). One can for example start from the Zeta alternating series (also called the Dirichlet eta function) η(s) ≡

∞ X (−1)n−1 , ns n=1

defining an analytic function for 0. When the complex number s satisfy 1, we have η(s) =

∞ ∞ X X 2 2 1 − = ζ(s) − s ζ(s). s s n (2n) 2 n=1 n=1

In other words, we have ζ(s) =

η(s) , 1 − 21−s

1.

(2)

Since η(s) is defined for 0, this identity (2) permits to define the Zeta function for all complex number s with positive real part, except for s = 1 for which we have a pole. The extension of the Zeta function to the domain