Constructive Approximation

, Volume 7, Issue 1, pp 389–399 | Cite as

The GHS inequality and the Riemann hypothesis

  • Charles M. Newman
Article

Abstract

LetV(t) be the even function on (−∞, ∞) which is related to the Riemann xi-function by Ξ(x/2)=4∫−∞ exp(ixtV(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 

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Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Charles M. Newman
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA

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