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Zeros of principal $$L$$-functions and random matrix theory. (English) Zbl 0866.11050
This paper studies the distribution of the distances between consecutive zeros of the Riemann zeta-function and of other primitive $$L$$-functions associated to cuspidal automorphic representations of $$\text{GL}_m$$ over $${\mathbb Q}$$. Write the zeros in the upper part of the critical strip as $${1\over2}+i\gamma_j$$, ordered with respect to the real part of $$\gamma_j$$, with multiplicities. The average distance between two consecutive numbers $$\tilde\gamma_j={m\over2\pi}\gamma_j\log |\gamma_j|$$ is one; this follows from the asymptotic distribution of the $$\gamma_j$$. The $$n$$-th level correlation between the first $$N$$ of these normalized zeros is described by the quantity $R_n(B_N,f) = {n!\over N} \sum_{\{x_1, \ldots , x_n\}\subset B_N} f(x_1,\ldots, x_n),$ where $$B_N=\{\tilde\gamma_j : 1\leq j\leq N\}$$, and where $$f$$ runs through test functions on $$\mathbb R^n$$ that are invariant under permutations of the coordinates and under translations in the direction $$(1,1,\ldots,1)$$.
The asymptotic behavior of $$R_n(B_N,f)$$ as $$N\rightarrow\infty$$ is given under assumptions concerning the $$L$$-function under consideration and concerning $$f$$. On the $$L$$-function one assumes the Riemann hypothesis and a mild technical condition (valid for $$m\leq 3$$). The test function $$f$$ has to have rapid decay in all directions in the hyperplane $$H$$ orthogonal to $$(1,1,\ldots,1)$$ and the support of its Fourier transform should be inside the region $$\sum_j |\xi_j|<2/m$$. As $$N\rightarrow\infty$$, the quantity $$R_n(B_N,f)$$ tends to the integral $\sqrt n \int_H f(y) W_n(y)\,dy$ over the hyperplane $$H$$. The density $$W_n(y)$$ is equal to the determinant of the $$n\times n$$-matrix with $$(i,j)$$-th element $$\sin\pi(y_i-y_j) /\pi(y_i-y_j)$$. This density has also turned up in random matrix theory.
This result is a consequence of the asymptotic behavior as $$T\rightarrow\infty$$ of the more complicated quantity $R_n(T,f,h) = \mathop{\textstyle\sum'}_{j_1,\ldots,j_n} h(\gamma_{j_1}/T) h(\gamma_{j_2}/T)\cdots h(\gamma_{j_n}/T) f(L\gamma_{j_1}/2\pi,\ldots, L\gamma_{j_n}/2\pi),$ where $$L=m\log T$$, and the sum runs over different indices. The auxiliary test function $$h$$ is the Fourier transform of a compactly supported smooth function. It is shown that $R_n(T,f,h) \sim {m\over 2\pi}T\log T \int_{-\infty}^\infty h(r)^n dr \sqrt n \int_{H} f(y) W_n(y)\,dy,$ without assumption of the Riemann hypothesis.
The proofs are long and complicated (at least to the reviewer). However, the authors do not restrict themselves to the technical details. At various places they indicate the ideas behind the computations.
A sum over $$n$$-tuples of zeros is obtained by an $$n$$-fold application of the explicit formula; this is where the auxiliary function $$h$$ enters the situation. In the resulting sum, the indices are not necessarily all distinct. It takes “combinatorial sieving” to go over to the sum $$R_n(T,f,h)$$.
In a separate section and in an appendix, information and some proofs are given concerning facts on automorphic $$L$$-functions needed in the paper.

##### MSC:
 11M50 Relations with random matrices 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11M41 Other Dirichlet series and zeta functions
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##### References:
 [1] L. Barthel and D. Ramakrishnan, A nonvanishing result for twists of $$L$$-functions of $$\mathrm GL(n)$$ , Duke Math. J. 74 (1994), no. 3, 681-700. · Zbl 0826.11022 [2] H. Davenport, Multiplicative Number Theory , 2nd ed., Graduate Texts in Math., vol. 74, Springer-Verlag, New York, 1980. · Zbl 0453.10002 [3] F. J. Dyson, Statistical theory of the energy levels of complex systems. III , J. Mathematical Phys. 3 (1962), 166-175. · Zbl 0105.41604 [4] I. M. Gelfand and D. Kazhdan, Representations of the group $$\mathrm GL(n,K)$$ 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. · Zbl 0348.22011 [5] R. Godement and H. Jacquet, Zeta Functions of Simple Algebras , Lecture Notes in Math., vol. 260, Springer-Verlag, Berlin, 1972. · Zbl 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). · Zbl 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. · Zbl 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. JSTOR: · Zbl 0525.22018 [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. · Zbl 0443.22013 [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. JSTOR: · Zbl 0473.12008 [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. · Zbl 0712.22011 [12] M. Kac, Toeplitz matrices, translation kernels, and a related problem in probability theory , Duke Math. J. 21 (1954), 501-509. · Zbl 0056.10201 [13] E. Landau, Über die Anzahl der Gitterpunkte in gewisser Bereichen, (Zweite Abhandlung) , Gött. Nach. (1915), 209-243. · JFM 45.0312.02 [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. · Zbl 0225.14022 [15] J. H. van Lint and R. M. Wilson, A Course in Combinatorics , Cambridge University Press, Cambridge, 1992. · Zbl 0769.05001 [16] W. Luo, Z. Rudnick, and P. Sarnak, On Selberg’s eigenvalue conjecture , Geom. Funct. Anal. 5 (1995), no. 2, 387-401. · Zbl 0844.11038 [17] W. Luo, Z. Rudnick, and P. Sarnak, On the “Ramanujan conjectures” for $$\mathrm GL(m)$$ , in preparation. · Zbl 0965.11023 [18] M. L. Mehta, Random Matrices , Academic Press, Boston, 1991. · Zbl 0780.60014 [19] C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $$\mathrm GL(n)$$ , Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605-674. · Zbl 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. · Zbl 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. JSTOR: · Zbl 0615.10049 [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. · Zbl 0329.20028 [24] I. I. Piatetski-Shapiro, Arithmetic Dirichlet series: conjectures, proceedings of conference in honor of G. Freiman, CIRM (Merseille, 1993) , [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. · Zbl 0837.11047 [27] R. Rumely, Numerical computations concerning the ERH , Math. Comp. 61 (1993), no. 203, 415-440, S17-S23. JSTOR: · Zbl 0792.11034 [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. · Zbl 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. · Zbl 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. JSTOR: · Zbl 0467.12013 [33] J. Shalika, The multiplicity one theorem for $$\mathrm GL\sbn$$ , Ann. of Math. (2) 100 (1974), 171-193. JSTOR: · Zbl 0316.12010 [34] F. Spitzer, A combinatorial lemma and its application to probability theory , Trans. Amer. Math. Soc. 82 (1956), 323-339. JSTOR: · Zbl 0071.13003
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