Moment inequalities and the Riemann hypothesis

Abstract

It is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann ξ-function are real. On writingξ(x/2)=8 ∫0∞ Φ(t) cos(xt)dt, it is known that a necessary condition that the Riemann hypothesis be valid is that the moments\(\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt}\) satisfy the Turan inequalities $$(\hat b_m (\lambda ))^2 > \left( {\frac{{2m - 1}}{{2m + 1}}} \right)\hat b_{m - 1} (\lambda )\hat b_{m + 1} (\lambda )(m \geqslant 1,\lambda \geqslant 0).$$ (*) We give here a constructive proof that log\(\Phi (\sqrt t )\) is strictly concave for 0 <t < ∞, and with this we deduce in Theorem 2.4 a general class of moment inequalities which, as a special case, establishes that the inequalities (*) are in fact valid for all real λ. As the case λ=0 of (*) corresponds to the Polya conjecture of 1927, this gives a new proof of the Polya conjecture.