Introduction

Chapter I. The pioneering work of Euler on the zeta function and Dirichlet's

subsequent introduction of L-functions provided a major conceptual advance in our

understanding of the set of all prime numbers. The identity

1 °° 1

-11

1 — p-s ~ Z^

n

s '

p n=l

which exhibits clearly the multiplicative and the additive properties of the ordinary

integers, is a cornerstone of the modern analytic theory of numbers, a theory which

evolved in the course of the last century into a vast and powerful enterprize. Within

this new framework, formal infinite products indexed by the primes, often called

"product formulas", are intended to be the carriers of both algebraic and geometric

as well as analytic and spectral information. The genesis of these ideas can be

traced back to Hilbert's formulation of his reciprocity law for the residue symbol,

as well as to the work done by Herbrand, Chevalley and Weil who forged the old

arithmetic concepts of unique factorization with the topological concepts present

in Tychonoff's Theorem on infinite products of locally compact spaces to produce

the basic tools of harmonic analysis on the groups of ideles and adeles.

On the analytic side, the study of zeta and L-functions dates back to one of

Riemann's three proofs of the functional equation for £(s) - the one based on the

transformation formula for the elliptic theta function:

(ThetaTransformation) ^

e

-(n+a)V* = ^J^ e~n2™+2™na.

nGZ neZ

To gain a better understanding of the origin of these ideas, we recall that a proof

of the theta transformation formula is based on the solution to the

following1.

"Initial value problem for heat conduction":

On a closed linearly extended heat conductor (a wire for instance)

of length 1, find a solution u(x,t) to the heat equation

uxx -ut = 0,

with continuous derivatives up to the second order for all values

of the variable x and for all t 0, having a prescribed set of

1See Courant-Hilbert [37]: Methods of Mathematical Physics , vol. II, p. 197

xiii