Multiple integral formulas for weighted zeta moments: the case of the sixth moment (Q6631621)

Let \(z\) be a complex number, \(d(n)\) denotes the number of positive divisors of an integer \(n\) and \(e(z) = e^{2\pi i z}\) with \(i^{2} =-1\). Let consider the three functions, the Riemann zeta function, the weighted one Eisenstein series and the autocorrelation function \(A\) which are defined by \N\[\N\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^{z}},\ \Re(z)>1,\N\]\N\[\NS_{0}(z)=\sum_{n=1}^{\infty}d(n)e(nz),\ \Im(z)>0\N\]\Nand \N\[\NA(z)=\int_{0}^{\infty}\left(\frac{1}{xz}-\frac{1}{e^{xz}-1}\right)\left(\frac{1}{x}-\frac{1}{e^{x}-1}\right)dx,\ \Re(z)>0.\N\]\NIn this paper under review, the authors prove in Theorem 1.1 exact formulas for \N\[\N\int_{-\infty}^{+\infty}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2k}\frac{e^{k(\pi-\delta)}}{\cosh(\pi t)^{k}}dt\N\]\N(weighted 2kth moments of the Riemann zeta function) for all integer \(k\geq 1\) and all \(\delta\in{(0,\pi)}\). In Theorem 1.2, they show a formula in the case \(k=3\) and \(\delta\in{(0,\pi/2)}\) which generalizes the ones that can be obtained for the 2nd and 4th moments and is a reformulation of Titchmarsh's results (see [\textit{E. C. Titchmarsh}, Proc. Lond. Math. Soc. (2) 27, 137--150 (1927; JFM 53.0313.02)]).