A note on the gaps between consecutive zeros of the Riemann zetafunction
Authors:
H. M. Bui, M. B. Milinovich and N. C. Ng
Journal:
Proc. Amer. Math. Soc. 138 (2010), 41674175
MSC (2010):
Primary 11M26; Secondary 11M06
Published electronically:
May 28, 2010
MathSciNet review:
2680043
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: Assuming the Riemann Hypothesis, we show that infinitely often consecutive nontrivial zeros of the Riemann zetafunction 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 zetafunction, 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), no. 4,
421–424. MR
749453 (86i:11048), 10.1112/blms/16.4.421
 3.
J.
B. Conrey, A.
Ghosh, and S.
M. Gonek, Large gaps between zeros of the zetafunction,
Mathematika 33 (1986), no. 2, 212–238 (1987).
MR 882495
(88g:11057), 10.1112/S0025579300011219
 4.
B.
Conrey and H.
Iwaniec, Spacing of zeros of Hecke 𝐿functions and the
class number problem, Acta Arith. 103 (2002),
no. 3, 259–312. MR 1905090
(2003h:11103), 10.4064/aa10335
 5.
R.
R. Hall, A new unconditional result about large spaces between zeta
zeros, Mathematika 52 (2005), no. 12,
101–113 (2006). MR 2261847
(2007g:11104), 10.1112/S0025579300000383
 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 zeros of the zeta
function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV,
St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence,
R.I., 1973, pp. 181–193. MR 0337821
(49 #2590)
 8.
H.
L. Montgomery and A.
M. Odlyzko, Gaps between zeros of the zeta function, Topics in
classical number theory, Vol. I, II (Budapest, 1981) Colloq. Math. Soc.
János Bolyai, vol. 34, NorthHolland, Amsterdam, 1984,
pp. 1079–1106. MR 781177
(86e:11072)
 9.
H.
L. Montgomery and P.
J. Weinberger, Notes on small class numbers, Acta Arith.
24 (1973/74), 529–542. Collection of articles
dedicated to Carl Ludwig Siegel on the occasion of his seventyfifth
birthday, V. MR
0357373 (50 #9841)
 10.
Julia
Mueller, On the difference between consecutive zeros of the Riemann
zeta function, J. Number Theory 14 (1982),
no. 3, 327–331. MR 660377
(83k:10074), 10.1016/0022314X(82)900671
 11.
Nathan
Ng, Large gaps between the zeros of the Riemann zeta function,
J. Number Theory 128 (2008), no. 3, 509–556. MR 2389854
(2008m:11176), 10.1016/j.jnt.2007.03.011
 12.
Atle
Selberg, The zetafunction and the Riemann hypothesis, C. R.
Dixième Congrès Math. Scandinaves 1946, Jul. Gjellerups
Forlag, Copenhagen, 1947, pp. 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)
 1.
 H. M. Bui, Large gaps between consecutive zeros of the Riemann zetafunction, 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), 421424. MR 749453 (86i:11048)
 3.
 J. B. Conrey, A. Ghosh, and S. M. Gonek, Large gaps between zeros of the zetafunction, Mathematika 33 (1986), 212238. 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, 259312. MR 1905090 (2003h:11103)
 5.
 R. R. Hall, A new unconditional result about large spaces between between zeta zeros, Mathematika 52 (2005), 101113. 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, 6584. 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, 181193. 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), NorthHolland, Amsterdam, 1984. MR 0781177 (86e:11072)
 9.
 H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, Acta Arith. 24 (1974), 529542. MR 0357373 (50:9841)
 10.
 J. Mueller, On the difference between consecutive zeros of the Riemann zetafunction, J. Number Theory 14 (1983), 393396. MR 660377 (83k:10074)
 11.
 N. Ng, Large gaps between the zeros of the Riemann zeta function, J. Number Theory 128 (2008), 509556. MR 2389854 (2008m:11176)
 12.
 A. Selberg, The zetafunction and the Riemann Hypothesis, Skandinaviske Matematikerkongres 10 (1946), 187200. 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, 4363. MR 0140732 (25:4146)
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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:
http://dx.doi.org/10.1090/S000299392010104434
Keywords:
Distribution of zeros,
Riemann Hypothesis,
Riemann zetafunction
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.
