On the values of a zeta function at nonpositive integers

Chong Hsio Fang and Minking Eie

Vol. 145 (1990), No. 2, 201–210

DOI: 10.2140/pjm.1990.145.201

Abstract

Let

be the zeta function associated with the principal Delaunay-Voronoi cone. A general theory asserts that ζ(s) has an analytic continuation which is holomorphic in the whole complex plane except possible poles at s = 3∕2, s = 1 and s = 1∕2. In this paper, we shall compute the values of ζ(s) at non-positive integers. It is not surprising to see that these values are rational numbers and can be expressed explicitly in terms of Bernoulli numbers; i.e.

Mathematical Subject Classification 2000

Primary: 11E45

Secondary: 11M41

Milestones

Received: 19 September 1988

Revised: 20 April 1989

Published: 1 October 1990