The Voronoi (summation) formula for GL(2) has long been a standard tool for studying analytic properties of automorphic forms and their L-functions.
There have been numerous results coming out the Voronoi formula on GL(2).
To Voronoy and his contemporaries, the formula appeared tailor-made to evaluate certain finite sums.
That seemed significant because several important questions in number theory involve finite sums of arithmetic quantities.
The former estimates the size of d(n), the number of positive divisors of an integer n. Dirichlet proved where
Gauss’ circle problem concerns the average size of for which Gauss gave the estimate Each problem has a geometric interpretation, with D(X) counting lattice points in the region
In the series of papers Voronoy developed geometric and analytic methods to improve both Dirichlet’s and Gauss’ bound.
Let ƒ be a Maass cusp form for the modular group PSL(2,Z) and a(n) its Fourier coefficients.