On a mean theorem for the second moment of the Riemann zeta-function (Q2910125)

As usual, let NEWLINE\[NEWLINE E(T) := \int_0^T|\zeta(\textstyle{\frac{1}{2}}+it)|^2\,dt - T\bigl(\log\frac{T}{2\pi} + 2\gamma-1\bigr) NEWLINE\]NEWLINE denote the error term in the mean theorem for the second moment of the Riemann zeta-function \(\zeta(s) = \sum_{n=1}^\infty n^{-s}\;(\Re s >1)\) on the critical line \(\Re s = 1/2\) (\(\gamma = -\Gamma'(1)\) is Euler's constant). An important problem is to determine the order of \(E(T)\). Applying a classical formula of \textit{F. V. Atkinson} for \(E(T)\) [Acta Math. 81, 353--376 (1949; Zbl 0036.18603)], \textit{D. R. Heath-Brown} [Mathematika 25, 177--184 (1979; Zbl 0387.10023)] showed that NEWLINE\[NEWLINE \int_0^P E^2(T)\,dT = cP^{3/2} + F(P), \; c = \frac{2}{3}(2\pi)^{-1/2}\sum_{n=1}^\infty d^2(n)n^{-3/2}, NEWLINE\]NEWLINE where \(d(n)\) is the number of divisors of \(n\) and \(F(T) = O(T^{5/4}\log^2T)\). This bound was later improved, and the best current result, due to \textit{Y.-K. Lau} and \textit{K.-M. Tsang} [Math. Proc. Camb. Philos. Soc. 146, No. 2, 277--287 (2009; Zbl 1229.11125)] is that NEWLINE\[NEWLINE F(T) = O(T\log^3T\log\log T).\tag{1} NEWLINE\]NEWLINE In the present interesting paper the authors obtain the mean value formula NEWLINE\[NEWLINE \int_0^PF(T)\,dT = -3\pi^{-2}P^2\log^2P\log\log P + O(P^2\log^2P).\tag{2} NEWLINE\]NEWLINE An immediate consequence of (2) is that \(F(T) = \Omega_-(T\log^2T\log\log T)\), so that the gap between this and the function standing in (1) is only a factor of \(\log T\). Also note that in (2) the error term is only by a factor of \(\log\log P\) smaller than the main term, so that the proof of (2) has to be carried out very carefully. This is indeed done successfully by the authors. They deduce (2) from a mean square formula for \(E(T)\) which contains a smooth function. To achieve this, they use a version of Atkinson's formula for \(E(T)\) due to \textit{T. Meurman} [Q. J. Math., Oxf. II. Ser. 38, 337--343 (1987; Zbl 0624.10032)], which also contains suitable weight functions, making the error terms small.