Mathematical Proceedings of the Cambridge Philosophical Society



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


ALEKSANDAR IVIC a1, KOHJI MATSUMOTO a2 and YOSHIO TANIGAWA a2
a1 Katedra Matematike RGF-a, Universitet u Beogradu, Djušina 7, 11000 Beograd, Serbia (Yugoslavia) e-mail: aleks@ivic.matf.bg.ac.yu, aivic@rgf.rgf.bg.ac.yu
a2 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan; e-mail: kohjimat@math.nagoya-u.ac.jp, tanigawa@math.nagoya-u.ac.jp

Abstract

We study Δ(x; φ), the error term in the asymptotic formula for [sum L: summation operator]n[less-than-or-eq, slant]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 L: summation operator]n[less-than-or-eq, slant]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.

(Received February 26 1998)
(Revised May 28 1998)