CONVOLUTIONS AND MEAN SQUARE ESTIMATES OF CERTAIN NUMBER-THEORETIC ERROR TERMS

Abstract

We study the convolution function C[f(x)]:=\int_1^x f(y)f\Bigl(\frac xy\Bigr)\frac{dy}y When f(x) is a suitable number-theoretic error term. Asymptotics and upper bounds for C[f(x)] are derived from mean square bounds for f(x). Some applications are given, in particular to |\zeta(\tfrac12+ix)|^{2k} and the classical Rankin--Selberg problem from analytic number theory.