Constructive Approximation

, Volume 7, Issue 1, pp 389–399

The GHS inequality and the Riemann hypothesis

  • Charles M. Newman

DOI: 10.1007/BF01888165

Cite this article as:
Newman, C.M. Constr. Approx (1991) 7: 389. doi:10.1007/BF01888165


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 11M26Secondary 60K3582A25

Key words and phrases

Riemann HypothesisGHS inequalityIsing modelLee-Yang theorem

Copyright information

© Springer-Verlag New York Inc 1991

Authors and Affiliations

  • Charles M. Newman
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA