Moments of Hardy's function over short intervals

Abstract

Let as usual $Z(t) = \zeta(1/2+it)\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\zeta(s) = \chi(s)\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is also proved that $$ H(\log T)^{k^2} \ll_{k,\alpha} \sum_{T<\gamma\le T+H}\max_{\gamma\le \tau_\gamma\le \gamma^+} |\zeta(1/2 + i\tau_\gamma)|^{2k} \ll_{k,\alpha} H(\log T)^{k^2}. $$ Here $k>1$ is a fixed integer, $\gamma, \gamma^+$ denote ordinates of consecutive complex zeros of $\zeta(s)$ and $T^\alpha \le H \le T$, where $\alpha$ is a fixed constant such that $0<\alpha \le 1$. This sharpens and generalizes a result of M.B. Milinovich.