Riemann’s Zeta-function and Dirichlet Series

Abstract

The function defined for s >1 by

$$\varsigma (s) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^s}}}}.$$

(4.1)

and now called Riemann’s zeta-function was first¹ considered seriously by L.Euler (1734/35,1740,1743,1748 Chap.15,1774) (see Stäckel² (1907/08)) who determined its value first³ at s = 2 (for an analysis of Euler’s arguments see McKinzie,Tuckey (1997)) and then at all even positive integers. He proved the following formula which expresses these values in terms of Bernoulli numbers:

$$\varsigma (2n) = \frac{{( - {1^{n - 1}}{B_{2n}})}}{{2(2n)!}}{(2\pi )^{2n}}.$$