On the Rankin-Selberg $L$-function related to the Godement-Jacquet $L$-function II

In this paper, we consider the $k$-th Riesz mean for the coefficients of the Rankin-Selberg $L$-function $L_{f\times f}(s)$ related to the Godement-Jacquet $L$-function with respect to $SL(n,\mathbb{Z})$. We establish an asymptotic formula for the $k$-th Riesz mean with an improved range and a better error term. As a result, we get an asymptotic relation for the partial sum of the coefficients of $L_{f\times f}(s)$.