Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the gaps between consecutive zeros of the Riemann zeta-function


Authors: H. M. Bui, M. B. Milinovich and N. C. Ng
Journal: Proc. Amer. Math. Soc. 138 (2010), 4167-4175
MSC (2010): Primary 11M26; Secondary 11M06
DOI: https://doi.org/10.1090/S0002-9939-2010-10443-4
Published electronically: May 28, 2010
MathSciNet review: 2680043
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.5155 times the average spacing and that infinitely often they differ by at least 2.6950 times the average spacing.


References [Enhancements On Off] (What's this?)

  • 1. H. M. Bui, Large gaps between consecutive zeros of the Riemann zeta-function, submitted. Available on the arXiv at http://arxiv.org/abs/0903.4007.
  • 2. J. B. Conrey, A. Ghosh, and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull. London Math. Soc. 16 (1984), 421-424. MR 749453 (86i:11048)
  • 3. J. B. Conrey, A. Ghosh, and S. M. Gonek, Large gaps between zeros of the zeta-function, Mathematika 33 (1986), 212-238. MR 882495 (88g:11057)
  • 4. J. B. Conrey and H. Iwaniec, Spacing of zeros of Hecke $ L$-functions and the class number problem, Acta Arith. 103 (2002), no. 3, 259-312. MR 1905090 (2003h:11103)
  • 5. R. R. Hall, A new unconditional result about large spaces between between zeta zeros, Mathematika 52 (2005), 101-113. MR 2261847 (2007g:11104)
  • 6. H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell System Tech. J. 40 1961, 65-84. MR 0140733 (25:4147)
  • 7. H. L. Montgomery, The pair correlation of the zeros of the zeta function, Proc. Symp. Pure Math. 24, A.M.S., Providence, RI, 1973, 181-193. MR 0337821 (49:2590)
  • 8. H. L. Montgomery and A. M. Odlyzko, Gaps between zeros of the zeta function, Colloq. Math. Soc. Jānos Bolyai, 34. Topics in Classical Number Theory (Budapest, 1981), North-Holland, Amsterdam, 1984. MR 0781177 (86e:11072)
  • 9. H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, Acta Arith. 24 (1974), 529-542. MR 0357373 (50:9841)
  • 10. J. Mueller, On the difference between consecutive zeros of the Riemann zeta-function, J. Number Theory 14 (1983), 393-396. MR 660377 (83k:10074)
  • 11. N. Ng, Large gaps between the zeros of the Riemann zeta function, J. Number Theory 128 (2008), 509-556. MR 2389854 (2008m:11176)
  • 12. A. Selberg, The zeta-function and the Riemann Hypothesis, Skandinaviske Matematikerkongres 10 (1946), 187-200. MR 0019676 (8:446i)
  • 13. D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty, I. Bell System Tech. J. 40 1961, 43-63. MR 0140732 (25:4146)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11M26, 11M06

Retrieve articles in all journals with MSC (2010): 11M26, 11M06


Additional Information

H. M. Bui
Affiliation: Mathematical Institute, University of Oxford, Oxford, OX1 3LB United Kingdom
Email: hung.bui@maths.ox.ac.uk

M. B. Milinovich
Affiliation: Department of Mathematics, University of Mississippi, University, Mississippi 38677
Email: mbmilino@olemiss.edu

N. C. Ng
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, Canada T1K 3M4
Email: ng@cs.uleth.ca

DOI: https://doi.org/10.1090/S0002-9939-2010-10443-4
Keywords: Distribution of zeros, Riemann Hypothesis, Riemann zeta-function
Received by editor(s): September 5, 2009
Received by editor(s) in revised form: February 9, 2010
Published electronically: May 28, 2010
Additional Notes: The first author was supported by an EPSRC Postdoctoral Fellowship
The second author was supported in part by a University of Mississippi College of Liberal Arts summer research grant
The third author was supported in part by an NSERC Discovery grant
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society