Introduction

The theory of compact Riemann surfaces brings together diverse areas of

mathematics. Its building blocks include vast areas of analysis (including

Lie theory), geometry/topology and algebra. This was our point of view

in our book on Riemann surfaces [6] and it dictated the material to be

included in that volume. In particular, we presented a modern approach

to the theory of compact Riemann surfaces based on classical methods that

prepared the reader to study the modern theories of moduli of surfaces.

In this book we head in a different direction and develop another classical

connection: to combinatorial number theory. We do not neglect, however,

the connections to the problem of uniformizing surfaces represented by very

special Fuchsian groups. Problems in number theory can be reformulated as

questions about Riemann surfaces, and many of the answers to some of these

questions are obtained using function theory. Even though it is an old idea to

use function theory (compact Riemann surfaces and automorphic forms) to

study analytic and combinatorial number theory and there are many results

in these fields, we found it hard to dig out the underlying function theory

in the publications of number theorists. No doubt, this is our failing. But

since others may also have a deficiency in this area, we decided to organize

the material from this point of view. There is new material in this book

that has not previously appeared in print, and part of our aim is to present

this material to as wide an audience as possible. Our more important aim,

however, is to expose the reader to a beautiful chapter in function theory

and its applications.

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