On Riesz means of the coefficients of the Rankin-Selberg series
Abstract
We study [Delta](x; [phi]), 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 [Delta](x; [phi]) to that of [Delta]1(x; [phi]), the error term of the weighted sum [sum L: summation operator]n[less-than-or-eq, slant]x(x[minus sign]n)cn. Then we prove an upper bound and a sharp mean square formula for [Delta]1(x; [phi]), 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 [Delta](x; [phi]). Some other related topics are also discussed.
- Publication:
-
Mathematical Proceedings of the Cambridge Philosophical Society
- Pub Date:
- July 1999
- DOI:
- 10.1017/S0305004199003564
- Bibcode:
- 1999MPCPS.127..117I