##
**Zeroes of zeta functions and symmetry.**
*(English)*
Zbl 0921.11047

Ten years ago, A. Odlyzko made what is considered by many specialists to be perhaps the most striking discovery – at the phenomenological level – about the zeta function since the work of Riemann. He found that the local spacings of the zeros obey the laws for the scaled spacings between the eigenvalues of a typical large unitary matrix, namely the laws of the Gaussian Unitary Ensemble (GUE). An impressive plot of the spacings \(\delta_j\), for \(10^{20}\leq j\leq 10^{20}+ 7\cdot 10^6\), fitting exactly the Gaudin distribution is the first figure in the paper, where a total of 11 plots are depicted.

The main aim of the authors is to contribute to the investigations that have been carried out by a number of groups during the last 20 years, trying to elucidate the suggestion by Hilbert and Polya, that there may exist a natural interpretation of the zeros of the Riemann zeta-function. The evidence is becoming quite convincing nowadays. Aside from the facts mentioned in the paragraph above, concerning the local spacing distributions of the high zeros of the Riemann zeta function (what can also be extended to many generalizations of it), it turns out that the low-lying zeros of various families of zeta functions follow laws corresponding to the eigenvalue distributions of members of the classical groups. Another point concerns the case of zeta functions of curves over finite fields and their generalizations, where a spectral interpretation of their zeros exists in terms of eigenvalues of Frobenius on cohomology.

In the paper, all these developments are reviewed, albeit not in a chronological order of discovery. As the authors themselves warn, they only concentrate on the line of work sketched above, ignoring the standard body of important work on zeta functions and \(L\)-functions that has been done with other purposes. The paper has 26 pages. The first section is devoted to the Montgomery-Odlyzko law, providing different plots for the distribution of nearest-neighbor spacings for the zeros of the Riemann zeta-function (already mentioned above), for the Ramanujan \(L\)-function and for the \(L\)-function associated to \(E\), and a plot of the pair correlation for the zeros of zeta based on \(8\cdot 10^6\) zeros near the \(10^{20}\)-th zero, versus the GUE conjectured density. Section 2 is devoted to the uses in this context of the random matrix models, which were introduced by Wigner in the 50’s, in an attempt at describing the resonance line of heavy nuclei. In section 3 they carry the analysis to the zeta functions corresponding to function fields, as introduced by Artin, and devote section 4 to the low lying zeros. Phenomenologically, it is found that the distribution of the low-lying zeros of certain families follow the laws dictated by symmetries associated with the family. Some applications are described in section 5, followed by a concluding section 6.

The final comment in the paper expresses the authors’ belief that for each \(L\)-function of a family, \(L(s,f)\), \(f\in{\mathcal F}\), there is a natural interpretation of the zeros of \(L(s,f)\) as the eigenvalues of an operator \(U(f)\) on some space \(H\). As \(f\) varies over \({\mathcal F}\) the \(U(f)\)’s would become equi-distributed in the space of such operators with a given symmetry type. In particular, the Riemann zeta-function sits in a family that has a symplectic symmetry, and thus the corresponding operator should preserve a symplectic form.

The main aim of the authors is to contribute to the investigations that have been carried out by a number of groups during the last 20 years, trying to elucidate the suggestion by Hilbert and Polya, that there may exist a natural interpretation of the zeros of the Riemann zeta-function. The evidence is becoming quite convincing nowadays. Aside from the facts mentioned in the paragraph above, concerning the local spacing distributions of the high zeros of the Riemann zeta function (what can also be extended to many generalizations of it), it turns out that the low-lying zeros of various families of zeta functions follow laws corresponding to the eigenvalue distributions of members of the classical groups. Another point concerns the case of zeta functions of curves over finite fields and their generalizations, where a spectral interpretation of their zeros exists in terms of eigenvalues of Frobenius on cohomology.

In the paper, all these developments are reviewed, albeit not in a chronological order of discovery. As the authors themselves warn, they only concentrate on the line of work sketched above, ignoring the standard body of important work on zeta functions and \(L\)-functions that has been done with other purposes. The paper has 26 pages. The first section is devoted to the Montgomery-Odlyzko law, providing different plots for the distribution of nearest-neighbor spacings for the zeros of the Riemann zeta-function (already mentioned above), for the Ramanujan \(L\)-function and for the \(L\)-function associated to \(E\), and a plot of the pair correlation for the zeros of zeta based on \(8\cdot 10^6\) zeros near the \(10^{20}\)-th zero, versus the GUE conjectured density. Section 2 is devoted to the uses in this context of the random matrix models, which were introduced by Wigner in the 50’s, in an attempt at describing the resonance line of heavy nuclei. In section 3 they carry the analysis to the zeta functions corresponding to function fields, as introduced by Artin, and devote section 4 to the low lying zeros. Phenomenologically, it is found that the distribution of the low-lying zeros of certain families follow the laws dictated by symmetries associated with the family. Some applications are described in section 5, followed by a concluding section 6.

The final comment in the paper expresses the authors’ belief that for each \(L\)-function of a family, \(L(s,f)\), \(f\in{\mathcal F}\), there is a natural interpretation of the zeros of \(L(s,f)\) as the eigenvalues of an operator \(U(f)\) on some space \(H\). As \(f\) varies over \({\mathcal F}\) the \(U(f)\)’s would become equi-distributed in the space of such operators with a given symmetry type. In particular, the Riemann zeta-function sits in a family that has a symplectic symmetry, and thus the corresponding operator should preserve a symplectic form.

Reviewer: E.Elizalde (Barcelona)

### MSC:

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

11M41 | Other Dirichlet series and zeta functions |

81Q99 | General mathematical topics and methods in quantum theory |

### Keywords:

spacing distributions of zeros; zeros of the Riemann zeta-function; zeta functions of curves over finite fields; Montgomery-Odlyzko law; Ramanujan \(L\)-function; pair correlation; random matrix models; symplectic symmetry
PDFBibTeX
XMLCite

\textit{N. M. Katz} and \textit{P. Sarnak}, Bull. Am. Math. Soc., New Ser. 36, No. 1, 1--26 (1999; Zbl 0921.11047)

Full Text:
DOI

### Online Encyclopedia of Integer Sequences:

Nearest integer to imaginary part of n-th zero of Riemann zeta function.### References:

[1] | A. Altland, M. Zirnbauer, “Non-Standard Symmetry Classes in Mesoscopic Normal-Super-Conducting Hybrid Structures,” Cond-Mat/9602137. |

[2] | E. Artin, “Quadratische Körper in Geibiet der Höheren Kongruzzen I and II,” Math. Zeit., 19, 153-296, (1924). |

[3] | E. Bombieri, Hilbert’s 8th problem: an analogue, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 269 – 274. Proc. Sympos. Pure Math., Vol. XXVIII. |

[4] | Armand Brumer, The rank of \?\(_{0}\)(\?), Astérisque 228 (1995), 3, 41 – 68. Columbia University Number Theory Seminar (New York, 1992). · Zbl 0851.11035 |

[5] | A. Brumer, R. Heath-Brown, “The Average Rank of Elliptic Curves II,” (preprint), (1992). |

[6] | E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros. I. Three- and four-point correlations, Nonlinearity 8 (1995), no. 6, 1115 – 1131. E. B. Bogomolny and J. P. Keating, Random matrix theory and the Riemann zeros. II. \?-point correlations, Nonlinearity 9 (1996), no. 4, 911 – 935. · Zbl 0898.60107 · doi:10.1088/0951-7715/9/4/006 |

[7] | Armand Brumer and Oisín McGuinness, The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 375 – 382. · Zbl 0741.14010 |

[8] | B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7 – 25. , https://doi.org/10.1515/crll.1963.212.7 B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79 – 108. · Zbl 0147.02506 · doi:10.1515/crll.1965.218.79 |

[9] | Alain Connes, Formule de trace en géométrie non-commutative et hypothèse de Riemann, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 12, 1231 – 1236 (French, with English and French summaries). · Zbl 0864.46042 |

[10] | Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. · Zbl 0453.10002 |

[11] | Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273 – 307 (French). Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137 – 252 (French). |

[12] | C. Denninger, “Evidence for a Cohomological Approach to Analytic Number Theory,” First European Math Congress, Vol. 1, (1992), 491-510, Birkhauser, (1994). |

[13] | W. Duke, J. Friedlander, H. Iwaniec, “Representations by the Determinant and Mean Values of \(L\)-Functions,” in Sieve Methods, Exponential Sums and their Applications in Number Theory, Cambridge University Press, 109-115, (1997). CMP 98:15 |

[14] | W. Duke, The critical order of vanishing of automorphic \?-functions with large level, Invent. Math. 119 (1995), no. 1, 165 – 174. · Zbl 0838.11035 · doi:10.1007/BF01245178 |

[15] | Freeman J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Mathematical Phys. 3 (1962), 166 – 175. · Zbl 0105.41604 · doi:10.1063/1.1703775 |

[16] | William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. · Zbl 0077.12201 |

[17] | M. Gaudin, “Sur la loi Limite de L’espacement de Valuers Propres D’une Matrics Aleatiore,” Nucl. Phys., 25, 447-458, (1961). · Zbl 0107.44605 |

[18] | Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108 – 118. · Zbl 0417.14031 |

[19] | F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), no. 1, 1 – 23. · Zbl 0725.11027 |

[20] | 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. · Zbl 0629.10032 |

[21] | Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of \?\?(2) and \?\?(3), Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471 – 542. · Zbl 0406.10022 |

[22] | M. L. Mehta and M. Gaudin, On the density of eigenvalues of a random matrix, Nuclear Phys. 18 (1960), 420 – 427. · Zbl 0107.35702 |

[23] | Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of \?-series, Invent. Math. 84 (1986), no. 2, 225 – 320. · Zbl 0608.14019 · doi:10.1007/BF01388809 |

[24] | D. R. Heath-Brown, The size of Selmer groups for the congruent number problem. II, Invent. Math. 118 (1994), no. 2, 331 – 370. With an appendix by P. Monsky. · Zbl 0815.11032 · doi:10.1007/BF01231536 |

[25] | Dennis A. Hejhal, On the triple correlation of zeros of the zeta function, Internat. Math. Res. Notices 7 (1994), 293ff., approx. 10 pp., issn=1073-7928, review=\MR{1283025}, doi=10.1155/S1073792894000334,. · Zbl 0813.11048 |

[26] | Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0451.53038 |

[27] | H. Iwaniec, “Topics in Analytic Number Theory,” Rutgers University Course, (1988). |

[28] | H. Iwaniec, W. Luo, P. Sarnak, “Low Lying Zeroes of Families of \(L\)-Functions,” (preprint), (1998). · Zbl 1012.11041 |

[29] | H. Iwaniec, P. Sarnak, “The Non-Vanishing of Central Values of Automorphic \(L\)-Functions and Siegel’s Zero,” (preprint), (1997). · Zbl 0992.11037 |

[30] | Hervé Jacquet, Principal \?-functions of the linear group, Automorphic forms, representations and \?-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 63 – 86. |

[31] | Nicholas M. Katz, An overview of Deligne’s work on Hilbert’s twenty-first problem, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 537 – 557. |

[32] | N. Katz, “Big Twists Have Big Monodromy” (in preparation), (1998). |

[33] | Nicholas M. Katz, Affine cohomological transforms, perversity, and monodromy, J. Amer. Math. Soc. 6 (1993), no. 1, 149 – 222. · Zbl 0815.14011 |

[34] | V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171 – 196 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1229 – 1253. |

[35] | E. Kowalski, P. Michel, “Sur de Rang de \(J_0 (N)\),” (preprint), (1997). |

[36] | E. Kowalski, P. Michel, “Sur les Zeros de Fonctions l Automorphes de Grand Niveau,” (preprint), (1997). |

[37] | N. Katz, P. Sarnak, “Random Matrices, Frobenius Eigenvalue and Monodromy,” AMS Colloq. series (to appear), (1999). · Zbl 0958.11004 |

[38] | N. Katz, P. Sarnak, “Zeroes of Zeta Functions, their Spaces and their Spectral Nature,” (1997 preprint version of the present paper). |

[39] | D. Zagier and G. Kramarz, Numerical investigations related to the \?-series of certain elliptic curves, J. Indian Math. Soc. (N.S.) 52 (1987), 51 – 69 (1988). · Zbl 0688.14016 |

[40] | R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 18 – 61. Lecture Notes in Math., Vol. 170. · Zbl 0225.14022 |

[41] | B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33 – 186 (1978). · Zbl 0394.14008 |

[42] | Madan Lal Mehta, Random matrices, 2nd ed., Academic Press, Inc., Boston, MA, 1991. · Zbl 0780.60014 |

[43] | Loïc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437 – 449 (French). · Zbl 0936.11037 · doi:10.1007/s002220050059 |

[44] | Hugh L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. · Zbl 0216.03501 |

[45] | 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. |

[46] | M. Ram Murty, The analytic rank of \?\(_{0}\)(\?)(\?), Number theory (Halifax, NS, 1994) CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 263 – 277. · Zbl 0851.11036 |

[47] | A. Odlyzko, “The \(10^{20}\)-th Zero of the Riemann Zeta Function and 70 Million of its Neighbors,” (preprint), A.T.T., (1989). |

[48] | Ali E. Özlük and C. Snyder, Small zeros of quadratic \?-functions, Bull. Austral. Math. Soc. 47 (1993), no. 2, 307 – 319. · Zbl 0777.11031 · doi:10.1017/S0004972700012545 |

[49] | A. Perelli and J. Pomykała, Averages of twisted elliptic \?-functions, Acta Arith. 80 (1997), no. 2, 149 – 163. · Zbl 0878.11022 |

[50] | R. S. Phillips and P. Sarnak, On cusp forms for co-finite subgroups of \?\?\?(2,\?), Invent. Math. 80 (1985), no. 2, 339 – 364. · Zbl 0558.10017 · doi:10.1007/BF01388610 |

[51] | B. Riemann, “Über die Anzahl der Primzahlen uter Einer Gegebenen Gröbe,” Montasb. der Berliner Akad., (1858160), 671-680. |

[52] | M. Rubinstein, “Evidence for a Spectral Interpretation of Zeros of \(L\)-Functions,” Thesis, Princeton University, (1998). · Zbl 1337.11058 |

[53] | Robert Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415 – 440, S17 – S23. · Zbl 0792.11034 |

[54] | Zeév Rudnick and Peter Sarnak, Zeros of principal \?-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269 – 322. A celebration of John F. Nash, Jr. · Zbl 0866.11050 · doi:10.1215/S0012-7094-96-08115-6 |

[55] | P. Sarnak, “\(L\)-Functions,” ICM Talk, Berlin, 1998. |

[56] | F.K. Schmidt, “Analytische Zahlen Theorie in Körpen der Charakteristik \(p\),” Math. Zeit., 33, 1-32, (1931). |

[57] | Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79 – 98. · Zbl 0311.10029 · doi:10.1112/plms/s3-31.1.79 |

[58] | Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440 – 481. · Zbl 0266.10022 · doi:10.2307/1970831 |

[59] | Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026 |

[60] | J. Silverman, “The Average Rank of a Family of Elliptic Curves,” (preprint), (1997). |

[61] | S. A. Stepanov, The number of points of a hyperelliptic curve over a finite prime field, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1171 – 1181 (Russian). · Zbl 0192.58002 |

[62] | J. Tate, “On the Conjecture of Birch Swinnerton-Dyer and a Geometric Analogue,” Seminar Bourbaki, Fev, (1966, Exp. 306). CMP 98:09 · Zbl 0199.55604 |

[63] | E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. · Zbl 0601.10026 |

[64] | J. Vanderkam, “The Rank of Quotients of \(J_0 (N)\),” (preprint), (1997). · Zbl 1013.11030 |

[65] | André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149 – 156 (German). · Zbl 0158.08601 · doi:10.1007/BF01361551 |

[66] | André Weil, Basic number theory, 3rd ed., Springer-Verlag, New York-Berlin, 1974. Die Grundlehren der Mathematischen Wissenschaften, Band 144. · Zbl 0326.12001 |

[67] | A. Weil, “Sur les Functions Algebriques à corps de Constantes Fini,” C.R.Acad. Sci., Paris, 210, 592-594, (1940). |

[68] | A. Weil, “On the Riemann Hypothesis in Function Fields,” Proc. Nat. Acad. Sci., U.S.A., 27, 345-349, (1941). · Zbl 0061.06406 |

[69] | E. Wigner, “Random Matrices in Physics,” Siam Review, 9, 1-23, (1967). · Zbl 0144.48202 |

[70] | S. Wong, “Rank Zero Twists of Elliptic Curves,” (preprint), (1996). |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.