The zeta function on the critical line: Numerical evidence for moments and random matrix theory models
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- by Ghaith A. Hiary and Andrew M. Odlyzko PDF
- Math. Comp. 81 (2012), 1723-1752 Request permission
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 $10^9$ zeros near zero number $10^{23}$. 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.
References
- 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
- M. V. Berry, Semiclassical formula for the number variance of the Riemann zeros, Nonlinearity 1 (1988), no. 3, 399–407. MR 955621
- M. V. Berry and J. P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev. 41 (1999), no. 2, 236–266. MR 1684543, DOI 10.1137/S0036144598347497
- V. Chandee, On the correlation of shifted values of the Riemann zeta function, arXiv:0910.0664v1 [math.NT].
- J. B. Conrey, A note on the fourth power moment of the Riemann zeta-function, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 225–230. MR 1399340
- J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of $L$-functions, Proc. London Math. Soc. (3) 91 (2005), no. 1, 33–104. MR 2149530, DOI 10.1112/S0024611504015175
- 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, DOI 10.1016/j.jnt.2007.05.013
- J. B. Conrey and A. Ghosh, On mean values of the zeta-function, Mathematika 31 (1984), no. 1, 159–161. MR 762188, DOI 10.1112/S0025579300010767
- J. B. Conrey and A. Ghosh, A conjecture for the sixth power moment of the Riemann zeta-function, Internat. Math. Res. Notices 15 (1998), 775–780. MR 1639551, DOI 10.1155/S1073792898000476
- J. B. Conrey and S. M. Gonek, High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), no. 3, 577–604. MR 1828303, DOI 10.1215/S0012-7094-01-10737-0
- Harold Davenport, Multiplicative number theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423
- N. G. de Bruijn, Asymptotic methods in analysis, 3rd ed., Dover Publications, Inc., New York, 1981. MR 671583
- Adrian Diaconu, Dorian Goldfeld, and Jeffrey Hoffstein, Multiple Dirichlet series and moments of zeta and $L$-functions, Compositio Math. 139 (2003), no. 3, 297–360. MR 2041614, DOI 10.1023/B:COMP.0000018137.38458.68
- Freeman J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Mathematical Phys. 3 (1962), 166–175. MR 143558, DOI 10.1063/1.1703775
- H. M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0466039
- S. M. Gonek, C. P. Hughes, and J. P. Keating, A hybrid Euler-Hadamard product for the Riemann zeta function, Duke Math. J. 136 (2007), no. 3, 507–549. MR 2309173, DOI 10.1215/S0012-7094-07-13634-2
- D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. London Math. Soc. (3) 38 (1979), no. 3, 385–422. MR 532980, DOI 10.1112/plms/s3-38.3.385
- G. A. Hiary and A. M. Odlyzko, book manuscript in preparation.
- C. P. Hughes, J. P. Keating, Private communications to G. A. Hiary.
- Aleksandar Ivić, On the fourth moment of the Riemann zeta-function, Publ. Inst. Math. (Beograd) (N.S.) 57(71) (1995), 101–110. Đuro Kurepa memorial volume. MR 1387359
- Aleksandar Ivić, The Riemann zeta-function, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the Riemann zeta-function with applications. MR 792089
- J. P. Keating and N. C. Snaith, Random matrix theory and $\zeta (1/2+it)$, Comm. Math. Phys. 214 (2000), no. 1, 57–89. MR 1794265, DOI 10.1007/s002200000261
- 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, DOI 10.1016/j.jnt.2010.05.008
- Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906
- 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
- www.netlib.org
- A. M. Odlyzko, The $10^{20}$-th zero of the Riemann zeta function and 175 million of its neighbors, www.dtc.umn.edu/$\sim$odlyzko
- A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308. MR 866115, DOI 10.1090/S0025-5718-1987-0866115-0
- A. M. Odlyzko, The $10^{22}$-nd 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, DOI 10.1090/conm/290/04578
- 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, DOI 10.1090/S0002-9947-1988-0961614-2
- 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, DOI 10.1017/CBO9780511550492.015
- “R manual”, http://cran.r-project.org/doc/manuals/R-intro.pdf.
- Atle Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), no. 5, 89–155. MR 20594
- E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
Additional Information
- Ghaith A. Hiary
- Affiliation: Pure Mathematics, University of Waterloo, 200 University Ave West, Waterloo, Ontario, Canada, N2L 3G1.
- MR Author ID: 930454
- Email: hiaryg@gmail.com
- Andrew M. Odlyzko
- Affiliation: Department of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, Minnesota 55455.
- MR Author ID: 133205
- Email: odlyzko@umn.edu
- 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. DMS-0757627 (FRG grant) and DMS-0635607. Computations were carried out at the Minnesota Supercomputing Institute.
- © Copyright 2011 American Mathematical Society
- Journal: Math. Comp. 81 (2012), 1723-1752
- MSC (2010): Primary 11M06, 11Y35, 11M50, 15B52
- DOI: https://doi.org/10.1090/S0025-5718-2011-02573-1
- MathSciNet review: 2904600