On some relations for Mellins transforms of Hardy's function (Q2910115)

Let NEWLINE\[NEWLINEM_{k}(s)=\int_1^{\infty}Z(x)^{k}x^{-s}\,dxNEWLINE\]NEWLINE be the modified Mellin transform of the \(k\)-th power of the Hardy function \(Z(x)\). The principle results of the paper are closed-form expressions for the mean-values NEWLINE\[NEWLINEI_k(\sigma):=\frac{1}{\pi}\int_0^\infty |M_{k}(\sigma+it)|^2\,dtNEWLINE\]NEWLINE when \(\sigma>1\) and \(k=1\) or \(2\). For example, when \(k=2\) one gets NEWLINE\[NEWLINEI_2(\sigma)=p\left(\frac{1}{\sigma-1}\right)+(2\sigma-1)\int_1^\infty E_2(x)x^{-2\sigma}\,dx,NEWLINE\]NEWLINE where \(p\) is a certain polynomial of degree 5, and \(E_2(t)\) is the error term in the asymptotic formula for the \(4\)-th power moment of \(\zeta(1/2+it)\). Note that the integral on the right converges for \(\sigma>3/4\).NEWLINENEWLINEFor \(k=1\) there is a similar expression involving NEWLINE\[NEWLINE\int_1^\infty G(x)x^{-1-2\sigma}\,dx,NEWLINE\]NEWLINE where \(G(x)\) is the error term in the asymptotic formula for the first power moment of \(E(t)\). In this case the integral involving \(G(x)\) converges for \(\sigma>3/8\).