Averages of shifted convolution sums for arithmetic functions

Abstract

Let f be a full-level cusp form for GL_m(\mathbb{Z} ) with Fourier coeffcients {{\displaystyle A}}_f({{\displaystyle c}}_{m-\mathrm{2}},\cdots ,{{\displaystyle c}}_{\mathrm{1}},n). Let \lambda (n) be either the von Mangoldt function \wedge (n) or the k-th divisor function {{\displaystyle \tau }}_k(n) : We consider averages of shifted convolution sums of the type : We succeed in obtaining a saving of an arbitrary power of the logarithm, provided that X^{\frac{\mathrm{8}}{\mathrm{33}}+\epsilon }\le H\le X^{\mathrm{1}-\epsilon } .

Key words： Average; shifted convolution sum; arithmetic function

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