MODULAR FUNCTIONS IN
ANALYTIC NUMBER THEORY

From the Preface: “An accurate (though uninspiring) title for this book would have been Applications of the Theory of the Modular Forms η(τ) and ϑ(τ)M to the Number-Theoretic functions p(n)p(n) and rs(n)rs(n) respectively. This is accurate because, except in the first two chapters, we deal exclusively with these two modular forms and these two number-theoretic functions. However, at the heart of these particular applications to the treatment of these specific number-theoretic functions lies the general theory of automorphic functions, a theory of far-reaching significance with important connections to a great many fields of mathematics. Indeed, together with Riemann surface theory, analytic number theory has provided the principal impetus for the development over the last century of the theory of automorphic functions ... I have tried to keep the book self-contained for those readers who have had a good first-year graduate course in analysis; and, in particular, I have assumed readers to be familiar with the Cauchy theory and the Lebesgue theorem of dominated convergence.”