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 $Z^{\prime }(t)$ are proved. It is also proved that $$H(\log T{)}^{k^2}{\ll }_{k,\alpha }\sum _{T<\gamma \le T+H}\underset{\gamma \le {\tau }_{\gamma }\le {\gamma }^{+}}{\max }|\zeta (1/2+i{\tau }_{\gamma }){|}^{2k}{\ll }_{k,\alpha }H(\log T{)}^{k^2}.$$ 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.

12 pages

References in corpus (2)

• Sharp conditional bounds for moments of the Riemann zeta function
• On some reasons for doubting the Riemann hypothesis