Geodesic
ON THE MEAN SQUARE OF THE RIEMANN ZETA-FUNCTION IN SHORT INTERVALS
Publications de l'Institut Mathématique, _N_S_85 (2009) no. 99, p. 1

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It is proved that, for $T^{\varepsilon}\leq G=G(T)\leq\frac12\sqrt{T}$, \begin{align*} ınt_T^{2T}\Bigl(I_1(t+G,G)-I_1(t,G)\Bigr)^2dt = TGum_{j=0}^3a_jłog^j\biggl(\frac{qrt{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{qrt{\pi}}ınt_{-ınfty}^ınfty |\z(frac12+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.
DOI : 10.2298/PIM0999001I
Classification : 11M06 11N37
Keywords: The Riemann zeta-function, the mean square in short intervals, upper bounds 
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     author = {Aleksandar Ivi\'c},
     title = {ON {THE} {MEAN} {SQUARE} {OF} {THE} {RIEMANN} {ZETA-FUNCTION} {IN} {SHORT} {INTERVALS}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {1 },
     publisher = {mathdoc},
     volume = {_N_S_85},
     number = {99},
     year = {2009},
     doi = {10.2298/PIM0999001I},
     zbl = {1224.11074},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM0999001I/}
}
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Aleksandar Ivić. ON THE MEAN SQUARE OF THE RIEMANN ZETA-FUNCTION IN SHORT INTERVALS. Publications de l'Institut Mathématique, _N_S_85 (2009) no. 99, p. 1 . doi: 10.2298/PIM0999001I

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