Averages of shifted convolution sums for arithmetic functions

摘要

Let f be a full-level cusp form for GL(m)(Z) with Fourier coefficients A(f)(c(m-2),..., c(1), n): Let (n) be either the von Mangoldt function (n) or the k-th divisor function (k)(n): We consider averages of shifted convolution sums of the type sigma(|h|H) | sigma(Xn2X)A(f) (1,..., 1, n+h)(n)|(2). We succeed in obtaining a saving of an arbitrary power of the logarithm, provided that X833+E H X1-epsilon.