On mean values of some zeta-functions in the critical strip

Ivić, Aleksandar

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 163-178.

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

$${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_k^2\left(n\right)n^{-2\sigma }T+R(k,\sigma ;T),$$

le terme erreur $R(k,\sigma ;T)$, pour lequel nous montrons de nouvelles bornes lorsque min $({\beta }_k,{\sigma }_k^{*})<\sigma <1$. 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.

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

$${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_k^2\left(n\right)n^{-2\sigma }T+R(k,\sigma ;T),$$

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

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