On the distribution of spacings between zeros of the zeta function

Author:
A. M. Odlyzko

Journal:
Math. Comp. **48** (1987), 273-308

MSC:
Primary 11M26; Secondary 11-04, 11Y35

MathSciNet review:
866115

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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 zeros and for zeros number to that are accurate to within , 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.

**[1]**R. A. Becker & J. M. Chambers,*S*:*An Interactive Environment for Data Analysis and Graphics*, Wadsworth, Belmont, Calif., 1984.**[2]**M. V. Berry,*Semiclassical theory of spectral rigidity*, Proc. Roy. Soc. London Ser. A**400**(1985), no. 1819, 229–251. MR**805089****[3]**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.)**[4]**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**, 10.1007/3-540-13392-5_1**[5]**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**, 10.1103/PhysRevLett.52.1**[6]**O. Bohigas, M. J. Giannoni & C. Schmit, "Spectral properties of the Laplacian and random matrix theories,"*J. Physique-Lettres*. (To appear.)**[7]**O. Bohigas, R. U. Haq & A. Pandey, "Higher-order correlations in spectra of complex systems,"*Phys. Rev. Lett.*, v. 54, 1985, pp. 1645-1648.**[8]**E. Bombieri & D. Hejhal, manuscript in preparation.**[9]**Richard P. Brent,*Algorithms for minimization without derivatives*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR**0339493****[10]**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**, 10.1090/S0025-5718-1979-0537983-2**[11]**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**, 10.1103/RevModPhys.53.385**[12]**W. S. Brown, "A simple but realistic model of floating-point computations,"*ACM Trans. Math. Software*, v. 7, 1981, pp. 445-480.**[13]**J. M. Chambers, W. S. Cleveland, B. Kleiner & P. A. Tukey,*Graphical Methods for Data Analysis*, Wadsworth, Belmont., Calif., 1983.**[14]**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**0310312****[15]**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**0328158****[16]**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**, 10.1093/qmath/36.1.43**[17]**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.**[18]**Cray Research, Inc.,*Cray X-MP and Cray-1 Computer Systems*;*Library Reference Manual SR*-0014, Revision I, Dec. 1984.**[19]**Cray Research, Inc.,*Cray-1 Computer Systems, S Series Mainframe Reference Manual HR*-0029, Nov. 1982.**[20]**D. Davies,*An approximate functional equation for Dirichlet 𝐿-functions*, Proc. Roy. Soc. Ser. A**284**(1965), 224–236. MR**0173352****[21]**Max Deuring,*Asymptotische Entwicklungen der Dirichletschen 𝐿-Reihen*, Math. Ann.**168**(1967), 1–30 (German). MR**0213309****[22]**Freeman J. Dyson,*Statistical theory of the energy levels of complex systems. II*, J. Mathematical Phys.**3**(1962), 157–165. MR**0143557****[23]**H. M. Edwards,*Riemann's Zeta Function*, Academic Press, New York, 1974.**[24]**David Freedman and Persi Diaconis,*On the histogram as a density estimator: 𝐿₂ theory*, Z. Wahrsch. Verw. Gebiete**57**(1981), no. 4, 453–476. MR**631370**, 10.1007/BF01025868**[25]**Akio Fujii,*On the zeros of Dirichlet 𝐿-functions. I*, Trans. Amer. Math. Soc.**196**(1974), 225–235. MR**0349603**, 10.1090/S0002-9947-1974-0349603-2**[26]**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**, 10.1515/crll.1978.302.167**[27]**Akio Fujii,*On the zeros of Dirichlet 𝐿-functions. II*, Trans. Amer. Math. Soc.**267**(1981), no. 1, 33–40. With corrections to: “On the zeros of Dirichlet 𝐿-functions. I” [Trans. Amer. Math. Soc. 196 (1974), 225–235; MR 50 #2096] and subsequent papers. MR**621970**, 10.1090/S0002-9947-1981-0621970-5**[28]**W. Gabcke,*Neue Herleitung und explizite Restabschätzung der Riemann-Siegel-Formel*, Ph. D. Dissertation, Göttingen, 1979.**[29]**P. X. Gallagher,*On the distribution of primes in short intervals*, Mathematika**23**(1976), no. 1, 4–9. MR**0409385****[30]**P. X. Gallagher,*Pair correlation of zeros of the zeta function*, J. Reine Angew. Math.**362**(1985), 72–86. MR**809967**, 10.1515/crll.1985.362.72**[31]**P. X. Gallagher and Julia H. Mueller,*Primes and zeros in short intervals*, J. Reine Angew. Math.**303/304**(1978), 205–220. MR**514680****[32]**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****[33]**A. Ghosh,*On the Riemann zeta function—mean value theorems and the distribution of \mid𝑆(𝑇)\mid*, J. Number Theory**17**(1983), no. 1, 93–102. MR**712972**, 10.1016/0022-314X(83)90010-0**[34]**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****[35]**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**, 10.1007/BF01475582**[36]**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, Birkhäuser Boston, Boston, MA, 1987, pp. 183–203. MR**1018376****[37]**S. M. Gonek,*A formula of Landau and mean values of 𝜁(𝑠)*, Topics in analytic number theory (Austin, Tex., 1982) Univ. Texas Press, Austin, TX, 1985, pp. 92–97. MR**804245****[38]**A. P. Guinand,*A summation formula in the theory of prime numbers*, Proc. London Math. Soc. (2)**50**(1948), 107–119. MR**0026086****[39]**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**, 10.1016/0167-2789(83)90138-0**[40]**Handbook of Mathematical Functions (M. Abramowitz and I. A. Stegun, eds.), National Bureau of Standards, Washington, D.C., 9th printing, 1970.**[41]**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.**[42]**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****[43]**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****[44]**D. Joyner, "Distribution theorems for*L*-functions." (To be published.)**[45]**D. Joyner, "On the Dyson-Montgomery hypothesis." (To be published.)**[46]**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**, 10.1007/BF02243438**[47]**M. G. Kendall & A. Stuart,*The Advanced Theory of Statistics*, 3rd ed., Hafner, New York, 1973.**[48]**Edmund Landau,*Über die Nullstellen der Zetafunktion*, Math. Ann.**71**(1912), no. 4, 548–564 (German). MR**1511674**, 10.1007/BF01456808**[49]**R. Sherman Lehman,*On the distribution of zeros of the Riemann zeta-function*, Proc. London Math. Soc. (3)**20**(1970), 303–320. MR**0258768****[50]**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.**[51]**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.**[52]**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**, 10.1090/S0025-5718-1986-0829637-3**[53]**MATHLAB Group,*MACSYMA Reference Manual*, MIT Laboratory for Computer Science, 1977.**[54]**Madan Lal Mehta,*Random matrices*, 2nd ed., Academic Press, Inc., Boston, MA, 1991. MR**1083764****[55]**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**0348823****[56]**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****[57]**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****[58]**Hugh L. Montgomery,*Extreme values of the Riemann zeta function*, Comment. Math. Helv.**52**(1977), no. 4, 511–518. MR**0460255****[59]**A. M. Odlyzko, "Distribution of zeros of the Riemann zeta function: Conjectures and computations." (Manuscript in preparation.)**[60]**A. M. Odlyzko and H. J. J. te Riele,*Disproof of the Mertens conjecture*, J. Reine Angew. Math.**357**(1985), 138–160. MR**783538**, 10.1515/crll.1985.357.138**[61]**A. M. Odlyzko & A. Schönhage, "Fast algorithms for multiple evaluations of the Riemann zeta function." (To be published.)**[62]**A. E. Ozluk,*Pair Correlation of Zeros of Dirichlet L-functions*, Ph. D. Dissertation, Univ. of Michigan, Ann Arbor, Mich., 1982.**[63]**C. E. Porter, ed.,*Statistical Theories of Spectra*:*Fluctuations*, Academic Press, New York, 1965.**[64]**N. L. Schryer,*A Test of a Computer's Floating-Point Arithmetic Unit*, AT & T Bell Laboratories Computing Science Technical Report #89, 1981.**[65]**N. L. Schryer, manuscript in preparation.**[66]**Atle Selberg,*Contributions to the theory of the Riemann zeta-function*, Arch. Math. Naturvid.**48**(1946), no. 5, 89–155. MR**0020594****[67]**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.**[68]**E. C. Titchmarsh,*The Theory of the Riemann Zeta-Function*, Oxford, at the Clarendon Press, 1951. MR**0046485****[69]**K.-M. Tsang,*The Distribution of the Values of the Riemann Zeta-function*, Ph. D. Dissertation, Princeton, 1984.**[70]**A. M. Turing,*A method for the calculation of the zeta-function*, Proc. London Math. Soc. (2)**48**(1943), 180–197. MR**0009612****[71]**A. L. Van Buren,*A Fortran Computer Program for Calculating the Linear Prolate Functions*, Report 7994, Naval Research Laboratory, Washington, May 1976.**[72]**André Weil,*Sur les “formules explicites” de la théorie des nombres premiers*, Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]**1952**(1952), no. Tome Supplementaire, 252–265 (French). MR**0053152****[73]**D. Winter & H. te Riele, Optimization of a program for the verification of the Riemann hypothesis,*Supercomputer*, v. 5, 1985, pp. 29-32.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1987-0866115-0

Article copyright:
© Copyright 1987
American Mathematical Society