On the asymptotics of the shifted sums of Hecke eigenvalue squares

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that forX2/3+epsilon < H < X(1-epsilon)there are constants B(h)such that n-expressionry sumexpressiontion lambda(f)(n)(2)lambda(f)(n+h)(2)-BhX=O-f,O-A,O-epsilon(X(logX)(-A))for all butO(f,A,epsilon)(H(logX)X <= n <= 2X(-3A))integersh is an element of[1,H]where{lambda(f)(n)}(n >= 1)are normalized Hecke eigenvalues of a fixedholomorphic cusp formf. Our method is based on the Hardy-Littlewood circle method. We divide the minorarcs into two partsm(1)andm(2). In order to treatm(2), we use the Hecke relations, a bound of Miller to apply somearguments from a paper of Matom & auml;ki, Radziwill and Tao. In order to treatm(1), we apply Parseval's identity andGallagher's lemma