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Periodica Mathematica Hungarica

, Volume 60, Issue 1, pp 37–40 | Cite as

A monotonicity property of Riemann’s xi function and a reformulation of the Riemann hypothesis

  • Jonathan Sondow
  • Cristian Dumitrescu
Article

Abstract

We prove that Riemann’s xi function is strictly increasing (respectively, strictly decreasing) in modulus along every horizontal half-line in any zero-free, open right (respectively, left) half-plane. A corollary is a reformulation of the Riemann Hypothesis.

Key words and phrases

critical line critical strip functional equation gamma function Hadamard product horizontal half-line open half-plane increasing in modulus monotonicity nonreal zero Riemann Hypothesis Riemann zeta function xi function 

Mathematics subject classification number

11M26 

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References

  1. [1]
    H. Davenport, Multiplicative Number Theory, 2nd ed., revised by H. L. Montgomery, Graduate Texts in Mathematics 74, Springer-Verlag, New York — Berlin, 1980.zbMATHGoogle Scholar
  2. [2]
    H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I, Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, Cambridge, 2007.zbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2010

Authors and Affiliations

  1. 1.New YorkUSA
  2. 2.KitchenerCanada

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