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Let be the error term in the mean square formula of the Riemann zeta-function in the critical strip . It is an analogue of the classical error term . The research of has a long history but the investigation of is quite new. In particular there is only a few information known about for . As an exploration, we study its mean value . In this paper, we give it an Atkinson-type series expansion and explore many of its properties as a function of .
We investigate the vertical version of the Sato-Tate conjecture for some GL₂ automorphic representations over totally real fields with specified local components at a finite set of finite places.
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