Some mean value results for the Riemann zeta-function (Q2710121)

In the paper under review the author obtains several interesting mean value results for the Riemann zeta-function \(\zeta(s)\). It contains evaluations of certain integrals involving i) the error term in the formula for the fourth moment of \(|\zeta(1/2+it)|\), and ii) \(|\zeta(1/2+it)|^{2k}\) with \(k=1\) or \(k=2\). Further, the author proves the asymptotic formula NEWLINE\[NEWLINE \int_1^T|\zeta(1/2+it)|^{2k}(\log(T/t))^\alpha dt=TP_{k,\alpha}(\log T)+O(T^{k/4+\epsilon}),NEWLINE\]NEWLINE where \(\alpha>1/2\) is fixed, \(k=1\) or \(k=2\), \(P_{k,\alpha}(z)\) is a polynomial of degree \(k^2\) in \(z\), and the implicit constant in the error term depends only on \(\alpha\). This formula for the logarithmic mean supports well known conjectures on the error terms in the asymptotic formulae for the second and fourth moment of the Riemann zeta-function on the critical line. Finally, a bound for the sum NEWLINE\[NEWLINE \sum_{r\leq R}\int_{t_r}^{t_r+G}|\zeta(1/2+it)|^{2k} dt \qquad(T^\epsilon\leq G\leq T^{1-\epsilon})NEWLINE\]NEWLINE over well-spaced points \(t_r\in[T,2T]\) is derived. This leads to the bound \(\ll T^{2+\epsilon}\) for the twelfth moment.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].