Geodesic
Convolutions and Mean Square Estimates of Certain Number-theoretic Error Terms
Publications de l'Institut Mathématique, _N_S_80 (2006) no. 94, p. 141

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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.
DOI : 10.2298/PIM0694141I
Classification : 11N37 11M06 44A15 26A12
Keywords: Convolution functions, slowly varying functions, the Riemann zeta-function, Dirichlet divisor problem, Abelian groups of a given order, the Rankin--Selberg problem 
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     author = {Aleksandar Ivi\'c},
     title = {Convolutions and {Mean} {Square} {Estimates} of {Certain} {Number-theoretic} {Error} {Terms}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {141 },
     publisher = {mathdoc},
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     year = {2006},
     doi = {10.2298/PIM0694141I},
     zbl = {1174.11076},
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     url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM0694141I/}
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Aleksandar Ivić. Convolutions and Mean Square Estimates of Certain Number-theoretic Error Terms. Publications de l'Institut Mathématique, _N_S_80 (2006) no. 94, p. 141 . doi: 10.2298/PIM0694141I

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