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 SondowEmail author
  • Cristian Dumitrescu


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



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations