Duke Mathematical Journal

Zeros of principal $L$-functions and random matrix theory

Zeév Rudnick and Peter Sarnak

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Article information

Source
Duke Math. J. Volume 81, Number 2 (1996), 269-322.

Dates
First available in Project Euclid: 19 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077245671

Mathematical Reviews number (MathSciNet)
MR1395406

Zentralblatt MATH identifier
0866.11050

Digital Object Identifier
doi:10.1215/S0012-7094-96-08115-6

Subjects
Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Citation

Rudnick, Zeév; Sarnak, Peter. Zeros of principal L -functions and random matrix theory. Duke Mathematical Journal 81 (1996), no. 2, 269--322. doi:10.1215/S0012-7094-96-08115-6. http://projecteuclid.org/euclid.dmj/1077245671.


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References

  • [1] L. Barthel and D. Ramakrishnan, A nonvanishing result for twists of $L$-functions of $\rm GL(n)$, Duke Math. J. 74 (1994), no. 3, 681–700.
  • [2] H. Davenport, Multiplicative Number Theory, 2nd ed., Graduate Texts in Math., vol. 74, Springer-Verlag, New York, 1980.
  • [3] F. J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Mathematical Phys. 3 (1962), 166–175.
  • [4] I. M. Gelfand and D. Kazhdan, Representations of the group $\rm GL(n,K)$ where $K$ is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 95–118.
  • [5] R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972.
  • [6] D. A. Hejhal, On the triple correlation of zeros of the zeta function, Internat. Math. Res. Notices (1994), no. 7, 293ff., approx. 10 pp. (electronic).
  • [7] H. Jacquet, Principal $L$-functions of the linear group, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 63–86.
  • [8] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464.
  • [9] H. Jacquet, I. I. Piatetski-Shapiro, and J. A. Shalika, Conducteur des représentations du groupe linéaire, Math. Ann. 256 (1981), no. 2, 199–214.
  • [10] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558.
  • [11] H. Jacquet and J. A. Shalika, Rankin-Selberg convolutions: Archimedean theory, Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 125–207.
  • [12] M. Kac, Toeplitz matrices, translation kernels, and a related problem in probability theory, Duke Math. J. 21 (1954), 501–509.
  • [13] E. Landau, Über die Anzahl der Gitterpunkte in gewisser Bereichen, (Zweite Abhandlung), Gött. Nach. (1915), 209–243.
  • [14] R. P. Langlands, Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin, 1970, pp. 18–61.
  • [15] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.
  • [16] W. Luo, Z. Rudnick, and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387–401.
  • [17] W. Luo, Z. Rudnick, and P. Sarnak, On the “Ramanujan conjectures” for $\mathrm GL(m)$, in preparation.
  • [18] M. L. Mehta, Random Matrices, Academic Press, Boston, 1991.
  • [19] C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\rm GL(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674.
  • [20] 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), Proc. Sympos. Pure Math., vol. 24, Amer. Math. Soc., Providence, 1973, pp. 181–193.
  • [21] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308.
  • [22] A. M. Odlyzko, The $10^20$ zero of the Riemann zeta function and $70$ million of its neighbors, preprint, A.T.&T., 1989.
  • [23] I. I. Piatetskii-Shapiro, Euler subgroups, Lie Groups and their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) ed. I. M. Gelfand, Halsted, New York, 1975, pp. 597–620.
  • [24] I. I. Piatetski-Shapiro, Arithmetic Dirichlet series: conjectures, proceedings of conference in honor of G. Freiman, CIRM (Merseille, 1993), to appear.
  • [25] B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Größe, Montasb. der Berliner Akad. (1858/60) 671–680; in Gessamelte Mathematische Werke, 2nd ed., Teubner, Leipzig, 1982, # VII.
  • [26] Z. Rudnick and P. Sarnak, The $n$-level correlations of zeros of the zeta function, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 10, 1027–1032.
  • [27] R. Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415–440, S17–S23.
  • [28] P. Sarnak, Course notes, Princeton University, 1995.
  • [29] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Collected Papers, Vol. 2, Springer-Verlag, Berlin, 1991, pp. 47–65.
  • [30] J.-P. Serre, Abelian $\ell$-adic Representations and Elliptic Curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, New York-Amsterdam, 1968.
  • [31] J.-P. Serre, 1981, Letter to J.-M. Deshouillers.
  • [32] F. Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355.
  • [33] J. Shalika, The multiplicity one theorem for $\rm GL\sbn$, Ann. of Math. (2) 100 (1974), 171–193.
  • [34] F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323–339.