The Atkinson formula near the critical line. II
This is the continuation of the author's previous work on the same subject [New Trends Probab. Stat. 2, 335-354 (1992; Zbl 0774.11048)]. The classical formula of \textit{F. V. Atkinson} [Acta Math. 81, 353-376 (1949; Zbl 0036.186)] gives an explicit expression for \(E(T)\), the error term in the mean square formula for \(\zeta ({1\over 2}+ it)\). The author is interested in the asymptotic behaviour of the quantities \[ \begin{aligned} E(T) &:= \int_0^T \bigl|\zeta (\sigma_T+ it) \bigr|^2 dt- T\log \Bigl( {\textstyle {T \over {2\pi}}} \Bigr)- (2\gamma -1)T,\\ E_T (\sigma_T) &:= \int^T_0 \bigl|\zeta (\sigma_T+ it) \bigr|^2 dt- T\zeta (1+ \delta_T)- (1- \delta_T )^{-1} (2\pi e)^{\delta_T} T^{1- \delta_T} \zeta (1- \delta_T), \end{aligned} \] where \(\sigma_T= {1\over 2}+ {1\over {l_T}}\), \(l_T> 0\), \(l_T\) tends monotonically to \(\infty\) and \(T\to \infty\), \(\delta_T= {2\over {l_T}}\). In this work the author complements his previous work and proves two results. In Theorem 1 it is shown that \(E_T\) has the same expression as Atkinson's formula (loc. cit.) for \(E(T)\) if \(l_T\gg T\log T\). In Theorem 2 he evaluates \(E_T\) directly when \(l_T\gg T\) and proves a formula, too complicated to be reproduced here, which is again similar to Atkinson's original formula for \(E(T)\) and which again contains an error term \(\ll \log^2 T\).
scientific article; zbMATH DE number 794491
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