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Let $E_sigma (T)$ 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 $E(T)$. The research of $E(T)$ has a long history but the investigation of $E_sigma (T)$ is quite new. In particular there is only a few information known about $E_sigma (T)$ for $3/4<sigma <1$. As an exploration, we study its mean value $int _1^TE_sigma (u),du$. In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of $T$

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