The zeta function on the critical line: Numerical evidence for moments and random matrix theory models
Authors:
Ghaith A. Hiary and Andrew M. Odlyzko
Journal:
Math. Comp. 81 (2012), 17231752
MSC (2010):
Primary 11M06, 11Y35, 11M50, 15B52
Published electronically:
December 19, 2011
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Abstract: Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for high moments and at large heights, the variability of moment values over adjacent intervals is substantial, even when those intervals are long, as long as a block containing zeros near zero number . More than anything else, the variability illustrates the limits of what one can learn about the zeta function from numerical evidence. It is shown that the rate of decline of extreme values of the moments is modeled relatively well by power laws. Also, some long range correlations in the values of the second moment, as well as asymptotic oscillations in the values of the shifted fourth moment, are found. The computations described here relied on several representations of the zeta function. The numerical comparison of their effectiveness that is presented is of independent interest, for future large scale computations.
 [AGZ]
Greg
W. Anderson, Alice
Guionnet, and Ofer
Zeitouni, An introduction to random matrices, Cambridge
Studies in Advanced Mathematics, vol. 118, Cambridge University Press,
Cambridge, 2010. MR 2760897
(2011m:60016)
 [B]
M.
V. Berry, Semiclassical formula for the number variance of the
Riemann zeros, Nonlinearity 1 (1988), no. 3,
399–407. MR
955621 (90e:81037)
 [BK]
M.
V. Berry and J.
P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM
Rev. 41 (1999), no. 2, 236–266 (electronic). MR 1684543
(2000f:11107), http://dx.doi.org/10.1137/S0036144598347497
 [Ch]
V. Chandee, On the correlation of shifted values of the Riemann zeta function, arXiv:0910.0664v1 [math.NT].
 [C]
J.
B. Conrey, A note on the fourth power moment of the Riemann
zetafunction, Analytic number theory, Vol. 1 (Allerton Park, IL,
1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA,
1996, pp. 225–230. MR 1399340
(97e:11096)
 [CFKRS1]
J.
B. Conrey, D.
W. Farmer, J.
P. Keating, M.
O. Rubinstein, and N.
C. Snaith, Integral moments of 𝐿functions, Proc.
London Math. Soc. (3) 91 (2005), no. 1, 33–104.
MR
2149530 (2006j:11120), http://dx.doi.org/10.1112/S0024611504015175
 [CFKRS2]
J.
B. Conrey, D.
W. Farmer, J.
P. Keating, M.
O. Rubinstein, and N.
C. Snaith, Lower order terms in the full moment conjecture for the
Riemann zeta function, J. Number Theory 128 (2008),
no. 6, 1516–1554. MR 2419176
(2009b:11139), http://dx.doi.org/10.1016/j.jnt.2007.05.013
 [CG1]
J.
B. Conrey and A.
Ghosh, On mean values of the zetafunction, Mathematika
31 (1984), no. 1, 159–161. MR 762188
(86a:11033), http://dx.doi.org/10.1112/S0025579300010767
 [CG2]
J.
B. Conrey and A.
Ghosh, A conjecture for the sixth power moment of the Riemann
zetafunction, Internat. Math. Res. Notices 15
(1998), 775–780. MR 1639551
(99h:11096), http://dx.doi.org/10.1155/S1073792898000476
 [CGo]
J.
B. Conrey and S.
M. Gonek, High moments of the Riemann zetafunction, Duke
Math. J. 107 (2001), no. 3, 577–604. MR 1828303
(2002b:11112), http://dx.doi.org/10.1215/S0012709401107370
 [D]
Harold
Davenport, Multiplicative number theory, 3rd ed., Graduate
Texts in Mathematics, vol. 74, SpringerVerlag, New York, 2000.
Revised and with a preface by Hugh L. Montgomery. MR 1790423
(2001f:11001)
 [DB]
N.
G. de Bruijn, Asymptotic methods in analysis, 3rd ed., Dover
Publications, Inc., New York, 1981. MR 671583
(83m:41028)
 [DGH]
Adrian
Diaconu, Dorian
Goldfeld, and Jeffrey
Hoffstein, Multiple Dirichlet series and moments of zeta and
𝐿functions, Compositio Math. 139 (2003),
no. 3, 297–360. MR 2041614
(2005a:11124), http://dx.doi.org/10.1023/B:COMP.0000018137.38458.68
 [Dy]
Freeman
J. Dyson, Statistical theory of the energy levels of complex
systems. III, J. Mathematical Phys. 3 (1962),
166–175. MR 0143558
(26 #1113)
 [E]
H.
M. Edwards, Riemann’s zeta function, Academic Press [A
subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1974. Pure and Applied Mathematics, Vol. 58. MR 0466039
(57 #5922)
 [GHK]
S.
M. Gonek, C.
P. Hughes, and J.
P. Keating, A hybrid EulerHadamard product for the Riemann zeta
function, Duke Math. J. 136 (2007), no. 3,
507–549. MR 2309173
(2008e:11100)
 [HB]
D.
R. HeathBrown, The fourth power moment of the Riemann zeta
function, Proc. London Math. Soc. (3) 38 (1979),
no. 3, 385–422. MR 532980
(81f:10052), http://dx.doi.org/10.1112/plms/s338.3.385
 [HO]
G. A. Hiary and A. M. Odlyzko, book manuscript in preparation.
 [H]
C. P. Hughes, J. P. Keating, Private communications to G. A. Hiary.
 [I1]
Aleksandar
Ivić, On the fourth moment of the Riemann
zetafunction, Publ. Inst. Math. (Beograd) (N.S.)
57(71) (1995), 101–110. Đuro Kurepa memorial
volume. MR
1387359 (97b:11109)
 [I2]
Aleksandar
Ivić, The Riemann zetafunction, A WileyInterscience
Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the
Riemann zetafunction with applications. MR 792089
(87d:11062)
 [KS]
J.
P. Keating and N.
C. Snaith, Random matrix theory and
𝜁(1/2+𝑖𝑡), Comm. Math. Phys.
214 (2000), no. 1, 57–89. MR 1794265
(2002c:11107), http://dx.doi.org/10.1007/s002200000261
 [Ko]
H.
Kösters, On the occurrence of the sine kernel in connection
with the shifted moments of the Riemann zeta function, J. Number
Theory 130 (2010), no. 11, 2596–2609. MR 2678864
(2011i:11144), http://dx.doi.org/10.1016/j.jnt.2010.05.008
 [M]
Madan
Lal Mehta, Random matrices, 3rd ed., Pure and Applied
Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam,
2004. MR
2129906 (2006b:82001)
 [Mo]
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)
 [Ne]
www.netlib.org
 [O1]
A. M. Odlyzko, The th zero of the Riemann zeta function and 175 million of its neighbors, www.dtc.umn.edu/odlyzko
 [O2]
A.
M. Odlyzko, On the distribution of spacings
between zeros of the zeta function, Math.
Comp. 48 (1987), no. 177, 273–308. MR 866115
(88d:11082), http://dx.doi.org/10.1090/S00255718198708661150
 [O3]
A.
M. Odlyzko, The 10²²nd zero of the Riemann zeta
function, Dynamical, spectral, and arithmetic zeta functions (San
Antonio, TX, 1999), Contemp. Math., vol. 290, Amer. Math. Soc.,
Providence, RI, 2001, pp. 139–144. MR 1868473
(2003h:11109), http://dx.doi.org/10.1090/conm/290/04578
 [OS]
A.
M. Odlyzko and A.
Schönhage, Fast algorithms for multiple
evaluations of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), no. 2, 797–809. MR 961614
(89j:11083), http://dx.doi.org/10.1090/S00029947198809616142
 [R]
Michael
Rubinstein, Computational methods and experiments in analytic
number theory, Recent perspectives in random matrix theory and number
theory, London Math. Soc. Lecture Note Ser., vol. 322, Cambridge
Univ. Press, Cambridge, 2005, pp. 425–506. MR 2166470
(2006d:11153), http://dx.doi.org/10.1017/CBO9780511550492.015
 [RMan]
``R manual'', http://cran.rproject.org/doc/manuals/Rintro.pdf.
 [S]
Atle
Selberg, Contributions to the theory of the Riemann
zetafunction, Arch. Math. Naturvid. 48 (1946),
no. 5, 89–155. MR 0020594
(8,567e)
 [T]
E.
C. Titchmarsh, The Theory of the Riemann ZetaFunction,
Oxford, at the Clarendon Press, 1951. MR 0046485
(13,741c)
 [AGZ]
 G. Anderson, A. Guionnet and O. Zeitouni, An Introduction to Random Matrices, Cambridge University press, 2009. MR 2760897
 [B]
 M. V. Berry, Semiclassical formula for the number variance of the Riemann zeros, Nonlinearity 1 (1988) 399407. MR 955621 (90e:81037)
 [BK]
 M. V. Berry, J. P. Keating, The Riemann zeros and eigenvalues asymptotics, SIAM Rev., vol. 41, no. 2, 1999, 236266. MR 1684543 (2000f:11107)
 [Ch]
 V. Chandee, On the correlation of shifted values of the Riemann zeta function, arXiv:0910.0664v1 [math.NT].
 [C]
 J. B. Conrey, A note on the fourth power moment of the Riemannzeta function, Analytic Number Theory, vol. 1, 225230, Progr. Math., 138, Birkhäuser, Boston, 1996. MR 1399340 (97e:11096)
 [CFKRS1]
 B. Conrey, D. Farmer, J. P. Keating, M. Rubinstein and N. Snaith, Integral moments of Lfunctions, Proc. London Math. Soc. (3) 91 (2005) 33104. MR 2149530 (2006j:11120)
 [CFKRS2]
 B. Conrey, D. Farmer, J. P. Keating, M. Rubinstein and N. Snaith, Lower order terms in the full moment conjecture for the Riemann zeta function, J. Num. Theory, Volume 128, Issue 6, June 2008, 15161554. MR 2419176 (2009b:11139)
 [CG1]
 J. B Conrey and A. Ghosh, Mean values of the Riemann zeta function, Mathematika 31 (1984) 159161. MR 762188 (86a:11033)
 [CG2]
 J. B Conrey and A. Ghosh, A conjecture for the sixth power moment of the Riemann zeta function, Int. Math. Res. Not., 15 (1998), 775780. MR 1639551 (99h:11096)
 [CGo]
 J. B. Conrey, S. M. Gonek, High moments of the Riemann zetafunction, Duke Math. J., v. 107, No. 3 (2001), 577604. MR 1828303 (2002b:11112)
 [D]
 H. Davenport, Multiplicative Number Theory, Springer, Third Edition, 2000. MR 1790423 (2001f:11001)
 [DB]
 N. G. de Bruijn, Asymptotic methods in analysis, Dover Publications, Dover ed., 1981. MR 671583 (83m:41028)
 [DGH]
 A. Diaconu, D. Goldfeld, J. Hoffstein, Multiple Dirichlet series and moments of zeta and functions, Composito Math. 139 (2003) 297360. MR 2041614 (2005a:11124)
 [Dy]
 F. Dyson, Statistical theory of the energy levels of complex systems. III, J. Math. Physics (Vol 3, no.1), JanuaryFebruary, 1962, 166175. MR 0143558 (26:1113)
 [E]
 Edwards, H. M., Riemann's Zeta Function, Academic Press, 1974. MR 0466039 (57:5922)
 [GHK]
 S. M. Gonek, C. P. Hughes, J. P. Keating, A hybrid EulerHadamard product for the Riemann zeta function, Duke Math. J. Volume 136, Number 3 (2007), 507549. MR 2309173 (2008e:11100)
 [HB]
 D. R. HeathBrown, The fourth power moment of the Riemann zeta function, Proc. London Math. Soc. (3), 38 (1979), 385422. MR 532980 (81f:10052)
 [HO]
 G. A. Hiary and A. M. Odlyzko, book manuscript in preparation.
 [H]
 C. P. Hughes, J. P. Keating, Private communications to G. A. Hiary.
 [I1]
 A. Ivić, On the fourth moment of the Riemann zetafunction, Publs. Inst. Math. (Belgrade) 57 (71) (1995), 101110. MR 1387359 (97b:11109)
 [I2]
 A. Ivić, The Riemann ZetaFunction, Wiley and Sons, 1985. MR 792089 (87d:11062)
 [KS]
 J. P. Keating, N. C. Snaith, Random matrix theory and , Commun. Math. Phys. 214 (2000), 5789. MR 1794265 (2002c:11107)
 [Ko]
 H. Kösters, On the occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function, J. Number Theory 130 (2010), no. 11, 25962609. MR 2678864 (2011i:11144)
 [M]
 M. Mehta, Random Matrices, Elsevier, 3rd Edition, 2004. MR 2129906 (2006b:82001)
 [Mo]
 H. Montgomery, The paircorrelation function for zeros of the zeta function, Proc. Symp. Pure Math., Vol. XXIV (1973), 181193. MR 0337821 (49:2590)
 [Ne]
 www.netlib.org
 [O1]
 A. M. Odlyzko, The th zero of the Riemann zeta function and 175 million of its neighbors, www.dtc.umn.edu/odlyzko
 [O2]
 A. M. Odlyzko, On the distribution of spacings between zeros of the Zeta function, Math. Comp., Vol. 48, No. 177 (1987), 273308. MR 866115 (88d:11082)
 [O3]
 A. M. Odlyzko, The nd zero of the Riemann zeta function, Dynamical, Spectral, and Arithmetic Zeta Functions, Amer. Math. Soc., Contemporary Math. series, no. 290 (2001), 139144. MR 1868473 (2003h:11109)
 [OS]
 A. M. Odlyzko and A. Schönhage, Fast algorithms for multiple evaluations of the Riemann Zeta function, Trans. Am. Math. Soc., Vol. 309, No. 2 (1988), 797809. MR 961614 (89j:11083)
 [R]
 M. Rubinstein, Computational methods and experiments in analytic number theory, Recent Perspectives in Random Matrix Theory and Number Theory, London Mathematical Society, 2005, 425506. MR 2166470 (2006d:11153)
 [RMan]
 ``R manual'', http://cran.rproject.org/doc/manuals/Rintro.pdf.
 [S]
 A. Selberg, Contributions to the theory of the Riemann zetafunction, Arch. Math. Naturvid. 48 (1946), 89155. MR 0020594 (8:567e)
 [T]
 E. Titchmarsh, The Theory of the Riemann Zetafunction, Oxford Science Publications, 2nd Edition, 1986. MR 0046485 (13:741c)
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Additional Information
Ghaith A. Hiary
Affiliation:
Pure Mathematics, University of Waterloo, 200 University Ave West, Waterloo, Ontario, Canada, N2L 3G1.
Email:
hiaryg@gmail.com
Andrew M. Odlyzko
Affiliation:
Department of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, Minnesota 55455.
Email:
odlyzko@umn.edu
DOI:
http://dx.doi.org/10.1090/S002557182011025731
PII:
S 00255718(2011)025731
Keywords:
Riemann zeta function,
moments,
OdlyzkoSchönhage algorithm
Received by editor(s):
August 12, 2010
Received by editor(s) in revised form:
May 18, 2011
Published electronically:
December 19, 2011
Additional Notes:
Preparation of this material was partially supported by the National Science Foundation under agreements No. DMS0757627 (FRG grant) and DMS0635607. Computations were carried out at the Minnesota Supercomputing Institute.
Article copyright:
© Copyright 2011
American Mathematical Society
