Abstract: The Poisson summation formula gives an exact formula for the sum of a function over the integers, namely as the sum of its Fourier transform over the integers. In the early 1900's Voronoi conjectured the existence of similar summation formulae for weighted sums of a(n) f(n), where a(n) is an arithmetic quantity. Voronoi himself worked out two examples, when a(n) is the number of divisors of n, and when a(n) is r_2(n), the number of ways to represent n as the sum of two squares. His formulae were used to reduce the error term in Gauss' circle problem (the number of lattice points in a circle of area X) from the trivial bound O(X^(1/2)), to O(X^(1/3)).

Although Voronoi proved his formulae by applying Poisson summation, they are nowadays thought of in terms of modular forms on the complex upper half plane, and L-functions. Generalizations to modular forms have been applied to a variety of problems in arithmetic and analysis. After describing the history, I will discuss a generalization of the Voronoi-type formulae to coefficients of automorphic forms on GL(n), and describe a recent application by Sarnak and Watson to bounding the L^4 norms of eigenfunctions.

(Joint work with Wilfried Schmid, Harvard University.)