On the mean square of the Riemann zeta-function in short intervals

TL;DR

This paper derives an asymptotic formula for the mean square of the difference of a smoothed version of the Riemann zeta-function over short intervals, providing explicit constants and error bounds.

Contribution

It introduces a new asymptotic formula for the mean square of the smoothed zeta-function differences in short intervals with explicit constants and error estimates.

Findings

Derived explicit asymptotic formula for the mean square of I_1(t+G)-I_1(t)

Provided bounds and constants for the asymptotic expression

Discussed generalizations to other differences and higher moments

Abstract

It is proved that, for $T^\epsilon\le G = G(T) \le {1\over2}\sqrt{T}$, $$ \int_T^{2T}\Bigl(I_1(t+G)-I_1(t)\Bigr)^2 dt = TG\sum_{j=0}^3a_j\log^j \Bigl({\sqrt{T}\over G}\Bigr) + O_\epsilon(T^{1+\epsilon}+ T^{1/2+\epsilon}G^2) $$ with some explicitly computable constants $a_j (a_3>0)$ where, for a fixed natural number $k$, $$I_k(t,G) = {1\over\sqrt{\pi}}\int_{-\infty}^\infty |\zeta(1/2+it+iu)|^{2k} {\rm e}^{-(u/G)^2} du. $$ The generalizations to the mean square of $I_1(t+U,G) - I_1(t,G)$ over $[T, T+H]$ and the estimation of the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed.