Values of Zeta Functions and Their Applications

Abstract

Zeta functions of various sorts are all-pervasive objects in modern number theory, and an ever-recurring theme is the role played by their special values at integral arguments, which are linked in mysterious ways to the underlying geometry and often seem to dictate the most important properties of the objects to which the zeta functions are associated. It is this latter property to which the word “applications” in the title refers. In this article we will give a highly idiosyncratic and prejudiced tour of a number of these “applications,” making no attempt to be systematic, but only to give a feel for some of the ways in which special values of zeta functions interrelate with other interesting mathematical questions. The prototypical zeta function is “Riemann’s” (math) and the prototypical result on special values is the theorem that ζ(k) = rational number × π^k (k > 0 even), (1) which Euler proved in 1735 and of which we will give a short proof in Section 1. (The “applications” in this case are the role which the rational numbers occurring on the right-hand side of this formula play in the theory of cyclotomic fields, in the construction of p-adic zeta functions, and in the investigation of Fermât’s Last Theorem.)