2

ALEXANDER FEL'SHTYN

Riemann's significant contribution here was his consideration in 1858 of the

zeta function as an analytic function.He first showed that the zeta function

has an analytic continuation to the complex plane as a meromorphic func-

tion with a single pole at s — 1 whose properties can on the one hand be

investigated by the techniques of complex analysis, and on the other yield

difficult theorems concerning the integers [73]. It is this connection between

the continuous and the discrete that is so wonderful. Riemann also showed

that the zeta function satisfied a functional equation of the form

C(l-a) = 7(*K(«)

where

1

{s) =

^2-s-T{s/2)/T((\-s)/2)

and T is the Euler gamma-function. In the course of his investigations Rie-

mann was led to suspect that all nontrivial zeros are on the line Re(s) = 1/2,

this is the Riemann Hypothesis which has been the central goal of research

to the present day. Although no proof has yet appeared various weak forms

of this conjecture, and in other contexts, analogues of it have been of con-

siderable significance.

0.1.2 Problems concerning zeta functions

Since the 19 th century, many special functions called zeta functions have

been defined and investigated. The main problems concerning zeta functions

are:

(I) Creation of new zeta functions.

(II) Investigation of the properties of zeta functions. Generally , zeta

functions have the following properties in common: 1) They are meromorphic

on the whole complex plane ; 2) they have Dirichlet series expansions ; 3) they

have Euler product expansions ; 4) they satisfy certain functional equations;

5) their special values play important role. Also, it is an important problem

to find the poles, residues, and zeros of zeta functions .

(III) Application to number theory, geometry, dynamical systems.

(IV) Study of the relations between different zeta functions.

Most of the functions called zeta functions or //-functions have the prop-

erties of problem (II).