There are incredibly rich connections between classical analysis and
number theory. For instance, analytic number theory contains many examples of
asymptotic expressions derived from estimates for analytic functions, such as
in the proof of the Prime Number Theorem. In combinatorial number theory, exact
formulas for number-theoretic quantities are derived from relations between
analytic functions. Elliptic functions, especially theta functions, are an
important class of such functions in this context, which had been made clear
already in Jacobi's Fundamenta nova. Theta functions are also
classically connected with Riemann surfaces and with the modular group
$\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which provide another path
for insights into number theory.
Farkas and Kra, well-known masters of the theory of Riemann surfaces
and the analysis of theta functions, uncover here interesting
combinatorial identities by means of the function theory on Riemann surfaces
related to the principal congruence subgroups $\Gamma(k)$. For
instance, the authors use this approach to derive congruences discovered by
Ramanujan for the partition function, with the main ingredient being the
construction of the same function in more than one way. The authors also obtain
a variant on Jacobi's famous result on the number of ways that an integer can
be represented as a sum of four squares, replacing the squares by triangular
numbers and, in the process, obtaining a cleaner result.
The recent trend of applying the ideas and methods of algebraic geometry to
the study of theta functions and number theory has resulted in great advances
in the area. However, the authors choose to stay with the classical point of
view. As a result, their statements and proofs are very concrete. In this
book the mathematician familiar with the algebraic geometry approach to theta
functions and number theory will find many interesting ideas as well as
detailed explanations and derivations of new and old results.
Highlights of the book include systematic studies of theta constant
identities, uniformizations of surfaces represented by subgroups of the modular
group, partition identities, and Fourier coefficients of automorphic
functions.
Prerequisites are a solid understanding of complex analysis, some
familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and
an interest in number theory. The book contains summaries of some of the
required material, particularly for theta functions and theta constants.
Readers will find here a careful exposition of a classical point of view of
analysis and number theory. Presented are numerous examples plus suggestions
for research-level problems. The text is suitable for a graduate course or for
independent reading.
Readership
Graduate students, research mathematicians interested in
complex analysis and number theory.