The moments of the function \(\Phi(t)\) (Q2713557)

Let NEWLINE\[NEWLINE \xi (x/2)=8\int^{\infty}_{0}\Phi (t)\cos (xt) dt, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Phi (t)=\pi e^{5t}\sum^{\infty}_{n=1}n^2(2n^2 \pi e^{4t}-3)e^{-\pi n^2 e^{4t}}, NEWLINE\]NEWLINE and set NEWLINE\[NEWLINEb_m=\int^{\infty}_{0}t^{2m} \Phi (t) dt,\qquad m=0,1,2,\ldots .NEWLINE\]NEWLINE \textit{G. Csordas, T. S. Norfolk} and \textit{R. S. Varga} [Trans. Am. Math. Soc. 296, 521-541 (1986; Zbl 0602.30030)] solved the moment problem of Pólya related to the Riemann hypothesis by proving Turán's inequalities NEWLINE\[NEWLINEb^2_m>{2m-1\over 2m+1} b_{m-1}b_{m+1},\qquad m=1,2,3,\ldots .NEWLINE\]NEWLINE They also announced that \(b_m\) is strictly decreasing for \(1\leq m\leq 339\) and strictly increasing for \(m\geq 339\), so that \(b_{339}\) is the minimum. The author gives the proofs for these facts here.