On the Atkinson formula for the $\zeta$ function

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).