ON THE MEAN SQUARE OF THE RIEMANN ZETA-FUNCTION IN SHORT INTERVALS

Aleksandar Ivic

Abstract: It is proved that, for $T^{\varepsilon}\leq G=G(T)\leq\frac12\sqrt{T}$, \begin{align*} \int_T^{2T}\Bigl(I_1(t+G,G)-I_1(t,G)\Bigr)^2dt &= TG\sum_{j=0}^3a_j\log^j\biggl(\frac{\sqrt{T}}{G}\biggr)
&\quad + O_\varepsilon(T^{1+\varepsilon}G^{1/2}+T^{1/2+\varepsilon}G^2) \end{align*} with some explicitly computable constants $a_j (a_3>0)$ where, for fixed $k\in\mathbb N$, $$ I_k(t,G)=\frac1{\sqrt{\pi}}\int_{-\infty}^\infty |\z(\tfrac12+it+iu)|^{2k}e^{-(u/G)^2}du. $$ 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.

Keywords: The Riemann zeta-function; the mean square in short intervals; upper bounds

Classification (MSC2000): 11M06; 11N37

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