Atkinson's formula for the mean square of $\zeta (s)$ with an explicit error term

Abstract

We provide an explicit $O\left({\log }^{2}T\right)$-term of the celebrated Atkinson's formula for the error term $E(T)$ of the second power moment of the Riemann zeta-function on the critical line. As an application, we obtain an explicit version of well-known estimate $E(T){\ll }_{\epsilon }{T}^{\frac{1}{3}+\epsilon }$.

42 pages, 1 figure, 2 tables

References in corpus (1)

• Error bounds for the large-argument asymptotic expansions of the Hankel and Bessel functions