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Moment inequalities and the Riemann hypothesis

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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 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.

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Communicated by E. B. Saff.

Dedicated to the memory of Peter Henrici, 1923–1987

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Csordas, G., Varga, R.S. Moment inequalities and the Riemann hypothesis. Constr. Approx 4, 175–198 (1988). https://doi.org/10.1007/BF02075457

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