On mean values of some zeta-functions in the critical strip

Abstract

For a fixed integer $k\ge 3$ and fixed $1/2<\sigma >1$ we consider $${\int }_1^T|\zeta (\sigma +it){|}^{2k}dt=\sum _{n=1}^{\infty }{d}_k^2(n)n^{-2\sigma }T+R(k,\sigma ;T),$$ where $R(k,\sigma ;T)=o(T)(T\to \infty )$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k,\sigma ;T)$ are given for certain ranges of $\sigma$. We also obtain new mean value results for the zeta-functions of holomorphic cusp forms and the Rankin-Selberg series.

To the memory of R.A. Rankin, 15 pages