Moment inequalities and the Riemann hypothesis |
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Authors: | George Csordas Richard S Varga |
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Institution: | 1. Department of Mathematics, University of Hawaii at Manoa, 96822, Honolulu, Hawaii, USA 2. Institute for Computational Mathematics, Kent State University, 44242, Kent, Ohio, USA
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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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