On the Atkinson formula for the $\zeta$ function

TL;DR

This paper refines and extends the Atkinson formula for the Riemann zeta function, providing explicit bounds and extending its applicability to a broader range of the complex plane using classical analytic number theory tools.

Contribution

It offers an explicit version of the Atkinson formula with improved bounds and extends its validity to a wider range of the complex plane for the zeta function.

Findings

Explicit bounds for the square mean integral of ζ on the critical line.

Extension of the Atkinson formula to Re(s) in [1/4, 3/4].

Improved bounds over previous results by Simonič and Helfgott et al.

Abstract

Thanks to Littlewood (1922) and Ingham (1928), we know the first two terms of the asymptotic formula for the square mean integral value of the Riemann zeta function $\zeta$ on the critical line. Later, Atkinson (1939) presented this formula with an error term of order $O(\sqrt{T}\log^{2}(T))$, which we call the Atkinson formula. Following the latter approach and the work of Titchmarsh (1986), we present an explicit version of the Atkinson formula, improving on a recent bound by Simoni\v{c} (2020). Moreover, we extend the Atkinson formula to the range $\Re(s)\in\left[\frac{1}{4},\frac{3}{4}\right]$, giving an explicit bound for the square mean integral value of $\zeta$ and improving on a bound by Helfgott and the authors (2019). We use mostly classical tools, such as the approximate functional equation and the explicit convexity bounds of the zeta function given by Backlund (1918).