A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip 3/4σ<1
Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 445-470.

Let E σ (T) be the error term in the mean square formula of the Riemann zeta-function in the critical strip 1/2<σ<1. It is an analogue of the classical error term E(T). The research of E(T) has a long history but the investigation of E σ (T) is quite new. In particular there is only a few information known about E σ (T) for 3/4<σ<1. As an exploration, we study its mean value 1 T E σ (u)du. In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of T.

Pour σ dans la bandre critique 1/2<σ<1, on note E σ (T) le terme d’erreur de la formule asymptotique de 1 T |ζ(σ+it)| 2 dt (pour T grand). C’est un analogue du terme d’erreur classique E(T) (=E 1/2 (T)). L’étude de E(T) a une longue histoire, mais celle de E σ (T) est assez récente. En particulier, lorsque 3/4<σ<1, on connaît peu d’informations sur E σ (T). Pour en gagner, nous étudions la moyenne 1 T E σ (u)du. Dans cet article, nous donnons une expression en série de type Atkinson et explorons quelques une des propriétés de la moyenne comme fonction en T.

Received:
Published online:
DOI: 10.5802/jtnb.553
Yuk-Kam Lau 1

1 Department of Mathematics The University of Hong Kong Pokfulam Road, Hong Kong
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Yuk-Kam Lau. A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma < 1$. Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 445-470. doi : 10.5802/jtnb.553. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.553/

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