A study of the mean value of the error term in the mean square formula of the Riemann zeta-function in the critical strip $3/4\le \sigma <1$

Lau, Yuk-Kam

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 2, pp. 445-470.

Pour $\sigma$ dans la bandre critique $1/2<\sigma <1$, on note $E_{\sigma }\left(T\right)$ le terme d’erreur de la formule asymptotique de ${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{2}dt$ (pour $T$ grand). C’est un analogue du terme d’erreur classique $E\left(T\right)$ ($=E_{1/2}\left(T\right)$). L’étude de $E\left(T\right)$ a une longue histoire, mais celle de $E_{\sigma }\left(T\right)$ est assez récente. En particulier, lorsque $3/4<\sigma <1$, on connaît peu d’informations sur $E_{\sigma }\left(T\right)$. Pour en gagner, nous étudions la moyenne ${\int }_1^T{E}_{\sigma }\left(u\right)du$. Dans cet article, nous donnons une expression en série de type Atkinson et explorons quelques une des propriétés de la moyenne comme fonction en $T$.

Let $E_{\sigma }\left(T\right)$ be the error term in the mean square formula of the Riemann zeta-function in the critical strip $1/2<\sigma <1$. It is an analogue of the classical error term $E\left(T\right)$. The research of $E\left(T\right)$ has a long history but the investigation of $E_{\sigma }\left(T\right)$ is quite new. In particular there is only a few information known about $E_{\sigma }\left(T\right)$ for $3/4<\sigma <1$. As an exploration, we study its mean value ${\int }_1^T{E}_{\sigma }\left(u\right)du$. In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of $T$.

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