On sums of squares of $|\zeta (\frac{1}{2}+i\gamma )|$ over short intervals

Abstract

A discussion involving the evaluation of the sum $$\sum_{T<\g\le T+H}|ζ(1/2+iγ)|^2$$ and some related integrals is presented, where $\gamma (>0)$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta (s)$. It is shown unconditionally that the above sum is $\ll H{\log }^{2}T\log \log T$ for $T^{2/3}{\log }^{4}T\ll H\le T$.

arXiv comment9 pages