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On Riesz means of the coefficients of the Rankin–Selberg series

Published online by Cambridge University Press:  01 July 1999

ALEKSANDAR IVIĆ ,
KOHJI MATSUMOTO  and
YOSHIO TANIGAWA
ALEKSANDAR IVIĆ
Affiliation:
Katedra Matematike RGF-a, Universitet u Beogradu, Djušina 7, 11000 Beograd, Serbia (Yugoslavia) e-mail: [email protected], [email protected]
KOHJI MATSUMOTO
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan; e-mail: [email protected], [email protected]
YOSHIO TANIGAWA
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan; e-mail: [email protected], [email protected]
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Abstract

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 127 , Issue 1 , July 1999 , pp. 117 - 131
Copyright
The Cambridge Philosophical Society 1999

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