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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 σ < 1

Yuk-Kam Lau — 2006

Journal de Théorie des Nombres de Bordeaux

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 ) d u . 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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