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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