The GHS inequality and the Riemann hypothesis Article

Received: 07 August 1989 Revised: 20 November 1989 DOI :
10.1007/BF01888165

Cite this article as: Newman, C.M. Constr. Approx (1991) 7: 389. doi:10.1007/BF01888165
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Abstract LetV (t ) be the even function on (−∞, ∞) which is related to the Riemann xi-function by Ξ(x /2)=4∫_{−∞} ^{∞} exp(ixt −V (t ))dt . In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV '(t )/t is increasing on (0, ∞). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV ' is convex on [0, ∞). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.

AMS classification Primary 11M26 Secondary 60K35 82A25

Key words and phrases Riemann Hypothesis GHS inequality Ising model Lee-Yang theorem Communicated by Richard Varga.

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Authors and Affiliations 1. Department of Mathematics University of Arizona Tucson USA 2. Courant Institute of Mathematical Sciences New York University New York USA