Double Series of Bessel Functions and the Circle and Divisor Problems

Abstract

This chapter is devoted to proofs of two remarkable identities involving infinite series of Bessel functions. One identity is associated with the classical circle problem, while the other is connected with the equally difficult divisor problem. Each of the identities is open to three different interpretations, with entirely different proofs needed for each interpretation. New methods of estimating trigonometric sums are introduced in our proofs.