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

TL;DR

This paper refines Atkinson's formula for the mean square of the Riemann zeta-function, providing explicit error bounds and improving the understanding of the error term in the second moment calculation.

Contribution

It offers an explicit error term for Atkinson's formula with a logarithmic bound, enhancing the precision of the second moment estimate of the zeta-function.

Findings

Explicit $O( ext{log}^2 T)$ error term for Atkinson's formula

Improved explicit bound for the error term $E(T)$

Enhanced understanding of the second moment of $ ext{zeta}(s)$

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 $E(T)\ll_{\varepsilon} T^{\frac{1}{3}+\varepsilon}$.