Duke Mathematical Journal

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

Zeév Rudnick and Peter Sarnak
Source: Duke Math. J. Volume 81, Number 2 (1996), 269-322.
First Page: Show Hide
Primary Subjects: 11M41
Secondary Subjects: 11F70
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

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

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.
Mathematical Reviews (MathSciNet): MR95d:11062
Zentralblatt MATH: 0826.11022
Digital Object Identifier: doi:10.1215/S0012-7094-94-07425-5
Project Euclid: euclid.dmj/1077288421
[2] H. Davenport, Multiplicative Number Theory, 2nd ed., Graduate Texts in Math., vol. 74, Springer-Verlag, New York, 1980.
Mathematical Reviews (MathSciNet): MR82m:10001
Zentralblatt MATH: 0453.10002
[3] F. J. Dyson, Statistical theory of the energy levels of complex systems. III, J. Mathematical Phys. 3 (1962), 166–175.
Mathematical Reviews (MathSciNet): MR26:1113
Zentralblatt MATH: 0105.41604
Digital Object Identifier: doi:10.1063/1.1703775
[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.
Mathematical Reviews (MathSciNet): MR53:8334
Zentralblatt MATH: 0348.22011
[5] R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972.
Mathematical Reviews (MathSciNet): MR49:7241
Zentralblatt MATH: 0244.12011
[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).
Mathematical Reviews (MathSciNet): MR96d:11093
Zentralblatt MATH: 0813.11048
[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.
Mathematical Reviews (MathSciNet): MR81f:22029
Zentralblatt MATH: 0413.12007
[8] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464.
Mathematical Reviews (MathSciNet): MR85g:11044
Zentralblatt MATH: 0525.22018
Digital Object Identifier: doi:10.2307/2374264
[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.
Mathematical Reviews (MathSciNet): MR83c:22025
Zentralblatt MATH: 0443.22013
Digital Object Identifier: doi:10.1007/BF01450798
[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.
Mathematical Reviews (MathSciNet): MR82m:10050a
Zentralblatt MATH: 0473.12008
Digital Object Identifier: doi:10.2307/2374103
[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.
Mathematical Reviews (MathSciNet): MR93d:22022
Zentralblatt MATH: 0712.22011
[12] M. Kac, Toeplitz matrices, translation kernels, and a related problem in probability theory, Duke Math. J. 21 (1954), 501–509.
Mathematical Reviews (MathSciNet): MR16,31a
Zentralblatt MATH: 0056.10201
Digital Object Identifier: doi:10.1215/S0012-7094-54-02149-3
Project Euclid: euclid.dmj/1077465879
[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.
Mathematical Reviews (MathSciNet): MR46:1758
Zentralblatt MATH: 0225.14022
Digital Object Identifier: doi:10.1007/BFb0079065
[15] J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.
Mathematical Reviews (MathSciNet): MR94g:05003
Zentralblatt MATH: 0769.05001
[16] W. Luo, Z. Rudnick, and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387–401.
Mathematical Reviews (MathSciNet): MR96h:11045
Zentralblatt MATH: 0844.11038
Digital Object Identifier: doi:10.1007/BF01895672
[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.
Mathematical Reviews (MathSciNet): MR92f:82002
Zentralblatt MATH: 0780.60014
[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.
Mathematical Reviews (MathSciNet): MR91b:22028
Zentralblatt MATH: 0696.10023
[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.
Mathematical Reviews (MathSciNet): MR49:2590
Zentralblatt MATH: 0268.10023
[21] A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp. 48 (1987), no. 177, 273–308.
Mathematical Reviews (MathSciNet): MR88d:11082
Zentralblatt MATH: 0615.10049
Digital Object Identifier: doi:10.2307/2007890
[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.
Mathematical Reviews (MathSciNet): MR53:10720
Zentralblatt MATH: 0329.20028
[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.
Mathematical Reviews (MathSciNet): MR96b:11124
Zentralblatt MATH: 0837.11047
[27] R. Rumely, Numerical computations concerning the ERH, Math. Comp. 61 (1993), no. 203, 415–440, S17–S23.
Mathematical Reviews (MathSciNet): MR94b:11085
Zentralblatt MATH: 0792.11034
Digital Object Identifier: doi:10.2307/2152965
[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.
Mathematical Reviews (MathSciNet): MR1220477
Zentralblatt MATH: 0787.11037
[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.
Mathematical Reviews (MathSciNet): MR41:8422
Zentralblatt MATH: 0186.25701
[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.
Mathematical Reviews (MathSciNet): MR82i:10030
Zentralblatt MATH: 0467.12013
Digital Object Identifier: doi:10.2307/2374219
[33] J. Shalika, The multiplicity one theorem for $\rm GL\sbn$, Ann. of Math. (2) 100 (1974), 171–193.
Mathematical Reviews (MathSciNet): MR50:545
Zentralblatt MATH: 0316.12010
Digital Object Identifier: doi:10.2307/1971071
[34] F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323–339.
Mathematical Reviews (MathSciNet): MR18,156e
Zentralblatt MATH: 0071.13003
Digital Object Identifier: doi:10.2307/1993051

2013 © Duke University Press

Duke Mathematical Journal

Duke Mathematical Journal

Turn MathJax Off
What is MathJax?