On the Atkinson formula for the \(\zeta\) function (Q2674842)

The authors derive new explicit bounds on the remainder terms in several classical formulae for the mean-square of the Riemann zeta function \(\zeta(s)\). Such formulae have been long-established but with implicit and unknown big-\(O\) constants. Specifically, on the critical line \(\Re (s)=1/2\), the formula for the mean-square of \(\zeta(s)\) takes the form \[ \int_0^T |\zeta(1/2+it)|^2\, dt = T\log T + (2\gamma-1-\log 2\pi))T+\mathcal{E}(T), \] where \(\gamma=0.5772\ldots\) is the Euler constant, and the remainder term \(\mathcal{E}(T)\) is a secondary term. It is known that \(\mathcal{E}(T)\) satisfies the theoretical upper bound \(\mathcal{E}(T)\ll T^{35/108+\varepsilon}\), \(35/108=0.32407\ldots\), and the current record upper bound on \(\mathcal{E}(T)\) is due to Bourgain and Watt, given by \(\mathcal{E}(T)\ll T^{1515/4816+\varepsilon}\), \(1515/4816=0.31457\ldots\). Furthermore, in the opposite direction, it is known that there exists a positive constant \(c\) such that \(|\mathcal{E}(T)| \ge c\,T^{1/4}\) for some arbitrarily large values of \(T\). Equivalently, \(\mathcal{E}(T)\) satisfies the \(\Omega\)-bound \(\mathcal{E}(T)=\Omega(T^{1/4})\). In the paper under review, the authors establish the explicit upper bound \[ |\mathcal{E}(T)|\le 18.169\, T^{1/2}\log^2 T,\quad (T\ge 100). \] This bound does not asymptotically (i.e., for very large \(T\)) match the theoretical big-\(O\) bounds mentioned earlier, which have \(T\) exponent less than \(1/2\), but is completely explicit involving no unknown big-\(O\) constants. Additionally, the authors establish an explicit bound on the remainder term in the mean-square of zeta on the line \(\Re{s}=\tau\), applicable for any \(\tau\ne 1/2\) such that \(1/4\le \tau\le 3/4\). For example, when \(1/2<\tau\le 3/4\), \[ \int_0^T|\zeta(\tau+it)|^2\,dt=\frac{(2\pi)^{2\tau-1}\zeta(2-2\tau)}{2-2\tau} T^{2-2\tau}+\mathcal{E}_{\tau}(T), \] and the authors show \[ |\mathcal{E}_{\tau}(T)|\le \frac{16.839}{(\tau-1/2)^2}\sqrt{T}\log^2 T,\qquad \left(T\ge 100\right). \] Lastly, the authors note that \textit{A. Simonič} and \textit{V. V. Starichkova} [J. Number Theory 244, 111--168 (2023; Zbl 07641101)] have very recently derived an explicit bound on the remainder \(\mathcal{E}(T)\) that is on the order of \(T^{\frac{1}{3}}\log^{\frac{5}{3}}T\). (They incorporated an explicit truncated Voronoi formula for the sum of the divisor function.) In particular, the exponent of \(T\) in this very recent result is \(1/3\), which is smaller than the exponent \(1/2\) in the authors' result. Nevertheless, the authors' bound still yields a sharper inequality at least up to \(T=10^{30}\).