On the asymptotics of the shifted sums of Hecke eigenvalue squares
Abstract
The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for X23+ε < H <X1-ε, there are constants Bh such that ΣX≤ n ≤ 2X λf(n)2λf(n+h)2-BhX=Of,A,ε(X ( X)-A) for all but Of,A,ε(H( X)-3A) integers h ∈ [1,H] where \λf(n)\n≥1 are normalized Hecke eigenvalues of a fixed holomorphic cusp form f. Our method is based on the Hardy-Littlewood circle method. We divide the minor arcs into two parts m1 and m2. In order to treat m2, we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matom\"aki, Radziwill and Tao. We apply Parseval's identity and Gallagher's lemma so as to treat m1.
0