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

For a fixed integer k≥3, and fixed 1 2<σ<1 we consider

∫ 1 T ζ(σ+it) 2k dt=∑ n=1 ∞ d k 2 (n)n -2σ T+R(k,σ;T),
where R(k,σ;T)=0(T)(T→∞) is the error term in the above asymptotic formula. Hitherto the sharpest bounds for R(k,σ;T) are derived in the range min (β k ,σ k * )<σ<1. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

Un entier k≥3 et un réel σ tel que 1 2<σ<1 étant fixés, on considère dans la formule asymptotique

∫ 1 T ζ(σ+it) 2k dt=∑ n=1 ∞ d k 2 (n)n -2σ T+R(k,σ;T),
le terme erreur R(k,σ;T), pour lequel nous montrons de nouvelles bornes lorsque min (β k ,σ k * )<σ<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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Aleksandar Ivić. On mean values of some zeta-functions in the critical strip. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 163-178. https://jtnb.centre-mersenne.org/item/JTNB_2003__15_1_163_0/

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