On the Kuznetsov formula

The Kuznetsov formula provides a deep connection between the spectral theory in hyperbolic Riemann surfaces and some exponential sums of arithmetic nature that has been extremely fruitful in modern number theory. Unfortunately the application of the Kuznetsov formula is by no means easy in practice because it involves oscillatory integral transforms with kernels given by special functions in non-standard ranges. In this paper we introduce a new formulation of the Kuznetsov formula that rules out these complications reducing the integral transforms to something almost as simple as a composition of two Fourier transforms. This formulation admits a surprisingly short and clean proof that does not require any knowledge about special functions, solving in this way another of the disadvantages of the classical approach. Moreover the reversed formula becomes more natural and in the negative case it reduces to a direct application of Fourier inversion. We also show that our approach is more convenient in applications and gives some freedom to play with explicit test functions.