On Riesz mean for the coefficients of twisted Rankin-Selberg L-functions

Abstract

Rankin obtained the asymptotic formula for the sum of coefficients of Rankin-Selberg $L$-series associated with a cusp form and a trivial character. $Ivi\acute{c}\text{-}$ Matsumoto-Tanigawa studied the error term in it by using a mean value formula which is yielded from the Voronoï formula of the Riesz mean. In this paper, we consider more general Rankin-Selberg $L$-series. It is associated with two cusp forms and a nontrivial character mod $d$. $Ivi\acute{c}$-Matsumoto-Tanigawa's method cannot be applied directly to our case. We consider the sum of coefficients of twisted Rankin-Selberg $L$-series by a modification of their method.