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

Ivić, Aleksandar

Publ. Inst. Math., Nouv. Sér. 85(99), 1-17 (2009).

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 \(T^\varepsilon\leq U=GT^{-\varepsilon}\ll T^{1/2-\varepsilon}\), \(U=U(T)\), \(G=G(T)\) \[ \int_T^{2T}\bigl(I_2(t+U,G)-I_2(t,G)\bigr)^2\,dt\ll_\varepsilon T^{2+\varepsilon}(U/G)^2. \]

Reviewer: Dragan Stankov (Beograd)

MSC:

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

Keywords:

mean square in short intervals; upper bounds

Citations:

Zbl 1226.11086