Zeros of principal \(L\)-functions and random matrix theory
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.
scientific article; zbMATH DE number 908676
scientific article