The Voronoi identity via the Laplace transform
The famous Voronoi identity is a series representation involving Bessel functions for the error term \(\Delta (x)\) in the asymptotic formula for the sum function of the divisor function \(d(n)\). Known proofs for this result are fairly complicated. The idea of the author is to show first that the Laplace transform of \(\Delta (x)\) and that of its asserted series expression coincide, and to deduce then that the original functions coincide as well. A virtue of this argument is its applicability to sum functions of other arithmetic functions such as \(r(n)\) (the number of representations of \(n\) as the sum of two squares) or the Fourier coefficients of a cusp form.
scientific article; zbMATH DE number 1203256