Transition mean values of shifted convolution sums

Let f   be a classical GL(2)GL(2) holomorphic cusp form of full level and even weight which is a normalized eigenfunction for the Hecke algebra, and let λ(n)λ(n) be its Fourier coefficients. In this paper we study “shifted convolution sums” ∑nλ(n)λ(n+h)∑nλ(n)λ(n+h) after averaging over many shifts h and obtain asymptotic estimates. The result is somewhat surprising: one encounters a transition region depending on the ratio of the square of the length of the average over h to the length of the shifted convolution sum. The phenomenon encountered in this paper is similar to that found by Conrey, Farmer and Soundararajan in their paper “Transition Mean Values of Real Characters”, and we discuss here the connection of both results to automorphic distributions, Eisenstein series and multiple Dirichlet series.