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

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

2003-11-23

arxiv.org/abs/math/0311403v1

Journal de Th\'eorie des Nombres de Bordeaux 15(2003), 163-178

Mathematics

Number Theory

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

Scientific paper

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.

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