Circle and divisor problems, and double series of Bessel functions

In approximately 1915, Ramanujan recorded two identities involving doubly infinite series of Bessel functions. The identities were brought to the mathematical public for the first time when his lost notebook was published in 1988, and are connected with the classical, long-standing circle and divisor problems, respectively. We provide a proof of the first identity for the first time by analytically continuing a new kind of Dirichlet series. Delicate estimates of exponential sums are needed, and the new methods we introduce may be of independent interest.