Convolutions and mean square estimates of certain number-theoretic error terms

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}...

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Veröffentlicht in:Publications de l'Institut mathématique (Belgrade) 2006, Vol.80 (94), p.141-156
1. Verfasser: Ivic, Aleksandar
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:0350-1302
1820-7405
DOI:10.2298/PIM0694141I