Moment inequalities and the Riemann hypothesis

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 Turán 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 Pólya conjecture of 1927, this gives a new proof of the Pólya conjecture.