Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the distribution of spacings between zeros of the zeta function
HTML articles powered by AMS MathViewer

by A. M. Odlyzko PDF
Math. Comp. 48 (1987), 273-308 Request permission

Abstract:

A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first ${10^5}$ zeros and for zeros number ${10^{12}} + 1$ to ${10^{12}} + {10^5}$ that are accurate to within $\pm {10^{ - 8}}$, and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave like eigenvalues of random Hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by relating them to the primes.
References
    R. A. Becker & J. M. Chambers, S: An Interactive Environment for Data Analysis and Graphics, Wadsworth, Belmont, Calif., 1984.
  • M. V. Berry, Semiclassical theory of spectral rigidity, Proc. Roy. Soc. London Ser. A 400 (1985), no. 1819, 229–251. MR 805089
    • M. V. Berry, "Riemann’s zeta function: A model for quantum chaos?," in Proc. Second Internat. Conf. on Quantum Chaos (T. Seligman, ed.), Springer-Verlag, Berlin and New York, 1986. (To appear.)
  • Oriol Bohigas and Marie-Joya Giannoni, Chaotic motion and random matrix theories, Mathematical and computational methods in nuclear physics (Granada, 1983) Lecture Notes in Phys., vol. 209, Springer, Berlin, 1984, pp. 1–99. MR 769113, DOI 10.1007/3-540-13392-5_{1}
  • O. Bohigas, M.-J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (1984), no. 1, 1–4. MR 730191, DOI 10.1103/PhysRevLett.52.1
    • O. Bohigas, M. J. Giannoni & C. Schmit, "Spectral properties of the Laplacian and random matrix theories," J. Physique-Lettres. (To appear.) O. Bohigas, R. U. Haq & A. Pandey, "Higher-order correlations in spectra of complex systems," Phys. Rev. Lett., v. 54, 1985, pp. 1645-1648. E. Bombieri & D. Hejhal, manuscript in preparation.
  • Richard P. Brent, Algorithms for minimization without derivatives, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0339493
  • Richard P. Brent, On the zeros of the Riemann zeta function in the critical strip, Math. Comp. 33 (1979), no. 148, 1361–1372. MR 537983, DOI 10.1090/S0025-5718-1979-0537983-2
  • T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Random-matrix physics: spectrum and strength fluctuations, Rev. Modern Phys. 53 (1981), no. 3, 385–479. MR 619406, DOI 10.1103/RevModPhys.53.385
    • W. S. Brown, "A simple but realistic model of floating-point computations," ACM Trans. Math. Software, v. 7, 1981, pp. 445-480. J. M. Chambers, W. S. Cleveland, B. Kleiner & P. A. Tukey, Graphical Methods for Data Analysis, Wadsworth, Belmont., Calif., 1983.
  • J. des Cloizeaux and M. L. Mehta, Some asymptotic expressions for prolate spheroidal functions and for the eigenvalues of differential and integral equations of which they are solutions, J. Mathematical Phys. 13 (1972), 1745–1754. MR 310312, DOI 10.1063/1.1665903
  • J. des Cloizeaux and M. L. Mehta, Asymptotic behavior of spacing distributions for the eigenvalues of random matrices, J. Mathematical Phys. 14 (1973), 1648–1650. MR 328158, DOI 10.1063/1.1666239
  • J. B. Conrey, A. Ghosh, D. Goldston, S. M. Gonek, and D. R. Heath-Brown, On the distribution of gaps between zeros of the zeta-function, Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 141, 43–51. MR 780348, DOI 10.1093/qmath/36.1.43
    • F. D. Crary & J. B. ROSSER, High Precision Coefficients Related to the Zeta Function, MRC Technical Summary Report #1344, Univ. of Wisconsin, Madison, May 1975, 171 pp.; reviewed by R. P. Brent in Math. Comp., v. 31, 1977, pp. 803-804. Cray Research, Inc., Cray X-MP and Cray-1 Computer Systems; Library Reference Manual SR-0014, Revision I, Dec. 1984. Cray Research, Inc., Cray-1 Computer Systems, S Series Mainframe Reference Manual HR-0029, Nov. 1982.
  • D. Davies, An approximate functional equation for Dirichlet $L$-functions, Proc. Roy. Soc. London Ser. A 284 (1965), 224–236. MR 173352, DOI 10.1098/rspa.1965.0060
  • Max Deuring, Asymptotische Entwicklungen der Dirichletschen $L$-Reihen, Math. Ann. 168 (1967), 1–30 (German). MR 213309, DOI 10.1007/BF01361542
  • Freeman J. Dyson, Statistical theory of the energy levels of complex systems. II, J. Mathematical Phys. 3 (1962), 157–165. MR 143557, DOI 10.1063/1.1703774
    • H. M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974.
  • David Freedman and Persi Diaconis, On the histogram as a density estimator: $L_{2}$ theory, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 4, 453–476. MR 631370, DOI 10.1007/BF01025868
  • Akio Fujii, On the zeros of Dirichlet $L$-functions. I, Trans. Amer. Math. Soc. 196 (1974), 225–235. MR 349603, DOI 10.1090/S0002-9947-1974-0349603-2
  • Akio Fujii, On the uniformity of the distribution of the zeros of the Riemann zeta function, J. Reine Angew. Math. 302 (1978), 167–205. MR 511699, DOI 10.1515/crll.1978.302.167
  • Akio Fujii, On the zeros of Dirichlet $L$-functions. II, Trans. Amer. Math. Soc. 267 (1981), no. 1, 33–40. MR 621970, DOI 10.1090/S0002-9947-1981-0621970-5
    • W. Gabcke, Neue Herleitung und explizite RestabschĂ€tzung der Riemann-Siegel-Formel, Ph. D. Dissertation, Göttingen, 1979.
  • P. X. Gallagher, On the distribution of primes in short intervals, Mathematika 23 (1976), no. 1, 4–9. MR 409385, DOI 10.1112/S0025579300016442
  • P. X. Gallagher, Pair correlation of zeros of the zeta function, J. Reine Angew. Math. 362 (1985), 72–86. MR 809967, DOI 10.1515/crll.1985.362.72
  • P. X. Gallagher and Julia H. Mueller, Primes and zeros in short intervals, J. Reine Angew. Math. 303(304) (1978), 205–220. MR 514680, DOI 10.1515/crll.1978.303-304.205
  • A. Ghosh, On Riemann’s zeta function—sign changes of $S(T)$, Recent progress in analytic number theory, Vol. 1 (Durham, 1979) Academic Press, London-New York, 1981, pp. 25–46. MR 637341
  • A. Ghosh, On the Riemann zeta function—mean value theorems and the distribution of $\mid S(T)\mid$, J. Number Theory 17 (1983), no. 1, 93–102. MR 712972, DOI 10.1016/0022-314X(83)90010-0
  • D. A. Goldston, Prime numbers and the pair correlation of zeros of the zeta-functions, Topics in analytic number theory (Austin, Tex., 1982) Univ. Texas Press, Austin, TX, 1985, pp. 82–91. MR 804244
  • D. R. Heath-Brown and D. A. Goldston, A note on the differences between consecutive primes, Math. Ann. 266 (1984), no. 3, 317–320. MR 730173, DOI 10.1007/BF01475582
  • Daniel A. Goldston and Hugh L. Montgomery, Pair correlation of zeros and primes in short intervals, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, BirkhĂ€user Boston, Boston, MA, 1987, pp. 183–203. MR 1018376
  • S. M. Gonek, A formula of Landau and mean values of $\zeta (s)$, Topics in analytic number theory (Austin, Tex., 1982) Univ. Texas Press, Austin, TX, 1985, pp. 92–97. MR 804245
  • A. P. Guinand, A summation formula in the theory of prime numbers, Proc. London Math. Soc. (2) 50 (1948), 107–119. MR 26086, DOI 10.1112/plms/s2-50.2.107
  • Martin C. Gutzwiller, Stochastic behavior in quantum scattering, Phys. D 7 (1983), no. 1-3, 341–355. Order in chaos (Los Alamos, N.M., 1982). MR 719062, DOI 10.1016/0167-2789(83)90138-0
    • Handbook of Mathematical Functions (M. Abramowitz and I. A. Stegun, eds.), National Bureau of Standards, Washington, D.C., 9th printing, 1970. R. U. Haq, A. Pandey & O. Bohigas, "Fluctuation properties of nuclear energy levels: Do theory and experiment agree?," Phys. Rev. Lett., v. 48, 1982, pp. 1086-1089.
  • D. R. Heath-Brown, Gaps between primes, and the pair correlation of zeros of the zeta function, Acta Arith. 41 (1982), no. 1, 85–99. MR 667711, DOI 10.4064/aa-41-1-85-99
  • 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
    • D. Joyner, "Distribution theorems for L-functions." (To be published.) D. Joyner, "On the Dyson-Montgomery hypothesis." (To be published.)
  • E. Karkoschka and P. Werner, Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion, Computing 27 (1981), no. 1, 57–69 (German, with English summary). MR 623176, DOI 10.1007/BF02243438
    • M. G. Kendall & A. Stuart, The Advanced Theory of Statistics, 3rd ed., Hafner, New York, 1973.
  • Edmund Landau, Über die Nullstellen der Zetafunktion, Math. Ann. 71 (1912), no. 4, 548–564 (German). MR 1511674, DOI 10.1007/BF01456808
  • R. Sherman Lehman, On the distribution of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 20 (1970), 303–320. MR 258768, DOI 10.1112/plms/s3-20.2.303
    • J. van de Lune, Some Observations Concerning the Zero-Curves of the Real and Imaginary Parts of Riemann’s Zeta Function, Report ZW 201/83, Mathematical Center, Amsterdam, December 1983. J. van de Lune, H. J. J. te Riele & D. T. Winter, Rigorous High Speed Separation of Zeros of Riemann’s Zeta Function, Report NW 113/81, Mathematical Center, Amsterdam, 1981.
  • J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), no. 174, 667–681. MR 829637, DOI 10.1090/S0025-5718-1986-0829637-3
    • MATHLAB Group, MACSYMA Reference Manual, MIT Laboratory for Computer Science, 1977.
  • Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR 1083764
  • M. L. Mehta and J. des Cloizeaux, The probabilities for several consecutive eigenvalues of a random matrix, Indian J. Pure Appl. Math. 3 (1972), no. 2, 329–351. MR 348823
  • 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
  • Hugh L. Montgomery, Distribution of the zeros of the Riemann zeta function, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 379–381. MR 0419378
  • Hugh L. Montgomery, Extreme values of the Riemann zeta function, Comment. Math. Helv. 52 (1977), no. 4, 511–518. MR 460255, DOI 10.1007/BF02567383
    • A. M. Odlyzko, "Distribution of zeros of the Riemann zeta function: Conjectures and computations." (Manuscript in preparation.)
  • A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. Reine Angew. Math. 357 (1985), 138–160. MR 783538, DOI 10.1515/crll.1985.357.138
    • A. M. Odlyzko & A. Schönhage, "Fast algorithms for multiple evaluations of the Riemann zeta function." (To be published.) A. E. Ozluk, Pair Correlation of Zeros of Dirichlet L-functions, Ph. D. Dissertation, Univ. of Michigan, Ann Arbor, Mich., 1982. C. E. Porter, ed., Statistical Theories of Spectra: Fluctuations, Academic Press, New York, 1965. N. L. Schryer, A Test of a Computer’s Floating-Point Arithmetic Unit, AT & T Bell Laboratories Computing Science Technical Report #89, 1981. N. L. Schryer, manuscript in preparation.
  • Atle Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), no. 5, 89–155. MR 20594
    • C. L. Siegel, "Über Riemanns Nachlass zur analytischen Zahlentheorie," Quellen und Studien zur Geschichte der Math. Astr. Phys., v. 2, 1932, pp. 45-80; reprinted in C. L. Siegel, Gesammelte Abhandlungen, vol. 1, Springer-Verlag, Berlin and New York, 1966, pp. 275-310.
  • E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
    • K.-M. Tsang, The Distribution of the Values of the Riemann Zeta-function, Ph. D. Dissertation, Princeton, 1984.
  • A. M. Turing, A method for the calculation of the zeta-function, Proc. London Math. Soc. (2) 48 (1943), 180–197. MR 9612, DOI 10.1112/plms/s2-48.1.180
    • A. L. Van Buren, A Fortran Computer Program for Calculating the Linear Prolate Functions, Report 7994, Naval Research Laboratory, Washington, May 1976.
  • AndrĂ© Weil, Sur les “formules explicites” de la thĂ©orie des nombres premiers, Comm. SĂ©m. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome SupplĂ©mentaire, 252–265 (French). MR 53152
    • D. Winter & H. te Riele, Optimization of a program for the verification of the Riemann hypothesis, Supercomputer, v. 5, 1985, pp. 29-32.
Similar Articles
Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 48 (1987), 273-308
  • MSC: Primary 11M26; Secondary 11-04, 11Y35
  • DOI: https://doi.org/10.1090/S0025-5718-1987-0866115-0
  • MathSciNet review: 866115