Superposition of zeros of distinct L-functions

In this paper we ®rst prove a weighted prime number theorem of an ``o ̈-diagonal'' type for Rankin-Selberg L-functions of automorphic representations of GLm and GLm 0 over Q. Then for m ˆ 1, or under the Selberg orthonormality conjecture for mV 2, we prove that nontrivial zeros of distinct primitive automorphic L-functions for GLm over Q are uncorrelated, for certain test functions whose Fourier transforms have restricted support. For the same test functions, we also prove that the n-level correlation of non-trivial zeros of a product of such L-functions follows the distribution of the superposition of GUE models for individual L-functions and GUEs of lower ranks. 1991 Mathematics Subject Classi®cation: 11F70, 11M26, 11M41. 1. Introduction. Rudnick and Sarnak [13] considered the n-level correlation of nontrivial zeros of a principal (primitive) L-function L…s; p† attached to an automorphic irreducible cuspidal representation p of GLm over Q. For a class of test functions with restricted support, they proved that the n-level correlation follows a GUE model of random matrix theory. This gives an evidence toward the conjectured Montgomery [9]-Odlyzko [10] [11] law. When the L-function is not principal, in particular, when L…s; p† is a product of several L-functions of lower ranks, the distribution of zeros was studied heuristically and numerically by Bogomolny and Leboeuf [1]. Their results suggest that zeros of a product of several principal L-functions follow the superposition of several GUEs. The goal of this article is to prove the superposition distribution of zeros of a product of several principal L-functions, for test functions with the same restricted support as in [13]. Our results indicate that the n-level correlation of non-trivial zeros of a non-principle L-function is the superposition of GUE models of individual Lfunction factors and products of lower rank GUEs. In other words, for an automorphic irreducible cuspidal representation p of GLm over a cyclic algebraic number Authors partially supported by National Natural Science Foundation (China) granta10041004, Trans-Century Training Programme Foundation for the Talents by the Ministry of Education (China), and National Science Foundation (USA) grant a DMS 97-01225. ®eld, the n-level correlation of non-trivial zeros of L…s; p† follows the structure of the base change liftings to p. Applications to distribution of primes will be given in subsequent papers. After introducing the notation and main theorems, we will prove an estimate of a sum associated with the Rankin-Selberg L-function L…s; p p 0† when ~ p l p 0n a for any real t, where a…g† ˆ jdet…g†j (Theorem 4.1). This result can be regarded as a weighted prime number theorem of an ``o ̈-diagonal'' type for the Rankin-Selberg Lfunction. Then following computation in §5 and §6, we will prove three main theorems on the zero correlation in §§7±9. 2. Notation. Let p be an automorphic irreducible cuspidal representation of GLm over Q with unitary central character. Denote by L…s; p† the L-function attached to p (see Jacquet [4] or [13] for de®nition). If we write p ˆNpUy pp, then L…s; p† ˆ Q p 3=2. Here L…s; py† ˆ Qm kˆ1 GR…s‡ mp…k††, where GR…s† ˆ pÿs=2G…s=2† and …mp…k†† is a set of m numbers associated to py. For p outside a ®nite set Sp of primes, pp is unrami®ed and the local factor L…s; pp† ˆ Qm kˆ1…1ÿap… p; k†pÿs†ÿ1, where ap…p; k†, k ˆ 1; . . . ;m, determine as eigenvalues a semisimple conjugacy class in GLm…C† associated to pp. For p A Sp, we can also write L…s; pp† in the above form by allowing some ap…p; k† to be zero. We note that our de®nition of L-functions contains the Riemann zeta function and Dirichlet L-functions. By a classical result of Godement and Jacquet [3], F…s; p† extends to an entire function with the exception of z…s†, which has a simple pole at s ˆ 1. F…s; p† also has a functional equation F…s; p† ˆ e…s; p†F…1ÿ s; ~ p†; where the automorphic irreducible cuspidal representation ~ p is the contragredient of p, and e…s; p† ˆ t…p†Qÿs p . Here Qp > 0 is the conductor of p (Jacquet, PiatetskiShapiro, and J. Shalika [5]), t…p† A C ;Q~ p ˆ Qp, and t…p†t…~ p† ˆ Qp. Denote ap…p† ˆ Pm kˆ1 ap…p; k† : Then a ~ p…p† ˆ ap… pl†. Set cp…n† ˆ L…n†ap…n†, where L…n† ˆ log p if n ˆ p and zero otherwise. Then for Re…s† > 3=2, we have L 0 L …s; p† ˆ ÿPy nˆ1 L…n†ap…n† ns : Note that for the Dirichlet L-function L…s; w† with w being a primitive character modulo q, we have m ˆ 1, Qp ˆ q, mw ˆ 0 if w…ÿ1† ˆ 1, and mw ˆ 1 if w…ÿ1† ˆ ÿ1. For paq, aw…p† ˆ w… p†, aw…p † ˆ w…p† ; for pjq, aw…p† ˆ 0, aw…p† ˆ 0. The contragredient of w is w. J. Liu, Y. Ye 420