Moment inequalities and the Riemann hypothesis (Q910518): Difference between revisions

The Riemann \(\xi\)-function can be expressed as a Fourier transform \[ \xi (x)=8\int^{\infty}_{0}\Phi (t)\cos xt dt=8\sum^{\infty}_{m=0}((- 1)^ m\hat b_ mx^{2m})/(2m)!, \] where the moments are given by \(\hat b_ m=\int^{\infty}_{0}t^{2m}\Phi (t)dt\) (m\(\geq 0)\), and \[ \Phi (t)=\sum^{\infty}_{n=1}(2n^ 4\pi^ 2e^{9t}-3n^ 2\pi e^{5t})\exp (-\pi n^ 2e^{4t}). \] The Riemann hypothesis is equivalent to the statement that \(\xi\) has only real zeros, while a necessary condition for the truth of the Riemann hypothesis is that the Turán inequalities \[ (\hat b_ m)^ 2>((2m-1)/(2m+1))\hat b_{m- 1}\hat b_{m+1} \] hold for \(m\geq 1\), a result now established [see a paper by the authors and \textit{T. S. Norfolk}, Trans. Am. Math. Soc. 296, 521-541 (1986; Zbl 0602.30030)]. In the paper under review, the authors consider a general transform of \(\Phi\) that includes, as a special case, the Fourier transform. They show that the analogues of the Turán inequalities, which also provide necessary conditions for the truth of the Riemann hypothesis, are true. Of primary importance is the result that log \(\Phi\) (\(\sqrt{t})\) is strictly concave on \(0<t<\infty\).