Publications de l'Institut Mathématique, Nouvelle Série Vol. 80(94), pp. 141–156 (2006) 

CONVOLUTIONS AND MEAN SQUARE ESTIMATES OF CERTAIN NUMBERTHEORETIC ERROR TERMSAleksandar IvicKatedra matematike RGF, Univerzitet u Beogradu, Beograd, SerbiaAbstract: 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 numbertheoretic 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. Keywords: Convolution functions, slowly varying functions, the Riemann zetafunction, Dirichlet divisor problem, Abelian groups of a given order, the Rankin–Selberg problem Classification (MSC2000): 11N37, 11M06, 44A15, 26A12 Full text of the article: (for faster download, first choose a mirror)
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