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.

**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 -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)**

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.