On mean values of some zeta-functions in the critical strip
Journal de théorie des nombres de Bordeaux, Volume 15 (2003) no. 1, pp. 163-178.

For a fixed integer $k\ge 3$, and fixed $\frac{1}{2}<\sigma <1$ we consider

 ${\int }_{1}^{T}{\left|\zeta \left(\sigma +it\right)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_{k}^{2}\left(n\right){n}^{-2\sigma }T+R\left(k,\sigma ;T\right),$
where $R\left(k,\sigma ;T\right)=0\left(T\right)\left(T\to \infty \right)$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R\left(k,\sigma ;T\right)$ are derived in the range min $\left({\beta }_{k},{\sigma }_{k}^{*}\right)<\sigma <1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

Un entier $k\ge 3$ et un réel $\sigma$ tel que $\frac{1}{2}<\sigma <1$ étant fixés, on considère dans la formule asymptotique

 ${\int }_{1}^{T}{\left|\zeta \left(\sigma +it\right)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_{k}^{2}\left(n\right){n}^{-2\sigma }T+R\left(k,\sigma ;T\right),$
le terme erreur $R\left(k,\sigma ;T\right)$, pour lequel nous montrons de nouvelles bornes lorsque min $\left({\beta }_{k},{\sigma }_{k}^{*}\right)<\sigma <1$. Nous obtenons également des majorations nouvelles pour les termes erreur dans le développement des moments d’ordre deux des fonctions zêta de formes paraboliques holomorphes et des séries de Rankin-Selberg.

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Ivić, Aleksandar. On mean values of some zeta-functions in the critical strip. Journal de théorie des nombres de Bordeaux, Volume 15 (2003) no. 1, pp. 163-178. http://www.numdam.org/item/JTNB_2003__15_1_163_0/

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