A remark on the Conrey-Ghosh theorem (Q1907495)

\textit{J. B. Conrey} and \textit{A. Ghosh} [Mathematika 31, 159-161 (1984; Zbl 0542.10034)] showed that if \(I_k (\sigma,T) = \int^T_1 |\zeta (\sigma + it) |^{2k} dt\), for real \(k \geq 0\), then \[ I_k (\textstyle {1 \over 2}, T) \geq \bigl\{ c_k + o(1) \bigr\} T(\log T)^{k^2} \] with an explicit constant \(c_k\), known to be optimal for \(k = 0\) and \(k = 1\). The present paper extends the result to \({1 \over 2} \leq \sigma \leq {1 \over 2} + o(1)\), with an appropriately modified lower bound.