## On the mean square of the Riemann zeta-function in short intervals.(English)Zbl 1224.11074

The author’s first theorem gives that, for $$T^\varepsilon\leq G=G(T)\leq\frac12\sqrt T$$, $\int_T^{2T}\bigl(I_1(t+G,G)-I_1(t,G)\bigr)^2\,dt= TG\sum_{j=0}^3a_j\log^j\biggl(\frac{\sqrt T}{G}\,\biggr) +O_{\varepsilon}(T^{1+\varepsilon}G^{1/2}+T^{1/2+\varepsilon}G^2),$ with some explicitly computable constants $$a_j$$ ($$a_3>0$$), where for fixed $$k\in\mathbb N$$, $I_k(t,G)=\frac1{\sqrt\pi}\int_{-\infty}^\infty| \zeta(1\!/2+it+iu)| ^{2k}e^{-(u/G)^2}\,du.$
In the proof the author uses some ideas from his previous investigations [Ramanujan J. 19, No. 2, 207–224 (2009; Zbl 1226.11086)], as well as results of Motohashi, Jutila and Atkinson. The generalization over short interval $$[T,T+H]$$ is estimated: $\int_T^{T+H}\bigl(I_1(t+U,G)-I_1(t,G)\bigr)^2\,dt\asymp\frac{HU^2}{G}\log^3\bigl(\sqrt T/G\bigr)\quad(T\to\infty).$
In the second theorem he proves that, for $$T^\varepsilon\leq U=GT^{-\varepsilon}\ll T^{1/2-\varepsilon}$$, $$U=U(T)$$, $$G=G(T)$$ $\int_T^{2T}\bigl(I_2(t+U,G)-I_2(t,G)\bigr)^2\,dt\ll_\varepsilon T^{2+\varepsilon}(U/G)^2.$

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11N37 Asymptotic results on arithmetic functions

### Keywords:

mean square in short intervals; upper bounds

Zbl 1226.11086
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