Some problems on mean values of the Riemann zeta-function
This paper begins by asking for which \(H,\alpha,\beta\) (depending on \(T)\) one has \[ \int_T^{T+H} \bigl|\zeta (\textstyle {1\over 2} + it) \bigr|^\alpha dt\leq \int_T^{T+H} \bigl|\zeta (\textstyle {1\over 2} + it) \bigr|^\beta dt. \] The author gives a partial answer and states various conjectures which would yield better results. Results and conjectures on the error term in the asymptotic formula for \(\int^T_0 |\zeta (\sigma + it) |^4dt\) for \(0< \sigma<1\) are then discussed. In particular it is shown that the integral is \[ {\zeta^4 (2\sigma) \over \zeta (4\sigma)} T+O(T^{2-2 \sigma} \log^3T). \] It is also asked whether one can find a \(\sigma<1\) for which \[ \int^T_0 \bigl|\zeta (\textstyle {1\over 2} + it) \bigr|^4 \bigl|\zeta (\sigma+it) \bigr|^2dt \ll_\varepsilon T^{1+ \varepsilon}. \] It seems to the reviewer that \(\sigma = 61/62\) is acceptable, since \(\zeta(\sigma+it)\) may then be approximated by a Dirichlet polynomial of length \(T^{1/5}\).
mean values of the Riemann zeta-function
scientific article; zbMATH DE number 912501