The \(n\)-level densities of low-lying zeros of quadratic Dirichlet \(L\)-functions (Q2859340)

This paper concerns the distribution of zeros close to the real axis, for quadratic Dirichlet \(L\)-functions. Assume the Generalized Riemann Hypothesis. Let \(f_1,\dots,f_n\) be even Schwartz functions, and define the corresponding \(n\)-level density for an individual quadratic Dirichlet \(L\)-function \(L(s,\chi_d)\) to be NEWLINE\[NEWLINED^{(n)}(d)=\sum_{\gamma_1,\dots,\gamma_n} f_1\left(\frac{\log |d|}{2\pi}\gamma_1\right)\cdots f_n\left(\frac{\log |d|}{2\pi}\gamma_n\right),NEWLINE\]NEWLINE where \(\gamma_1,\dots,\gamma_n\) are ordinates of zeros \(\tfrac{1}{2}+i\gamma\) of \(L(s,\chi_d)\) subject to the condition that \(\gamma_j\not=\pm\gamma_k\) for \(j\not=k\).NEWLINENEWLINEThe Katz-Sarnak density conjecture predicts that the average of \(D^{(n)}(d)\) tends to the same limit as one would have for normalized eigenvalues near 1 of the symplectic group of order \(N\), as \(N\) tends to infinity. This was shown by \textit{M. Rubinstein} [Duke Math. J. 109, No. 1, 147--181 (2001; Zbl 1014.11050)] when the product \(f_1\cdots f_n\) is replaced by an \(n\)-variable function whose Fourier transform is supported in \(\sum_1^n|u_i|<1\). \textit{P. Gao} [\(n\)-level density of the low-lying zeros of quadratic Dirichlet \(L\)-functions. PhD thesis, Michigan (2005)] considered the case of products \(f_1\cdots f_n\) as above, and handled the situation in which NEWLINE\[NEWLINE\prod_{i=1}^n \text{supp}(\hat{f_i})\subseteq\{(u_1,\dots,u_n)\in\mathbb{R}^n: \sum_{i=1}^n|u_i|<2\}.\tag{\(*\)} NEWLINE\]NEWLINE Although Gao was able to find the resulting limit, he was only able to show that it agreed with the Katz-Sarnak conjecture in the case \(n\leq 3\).NEWLINENEWLINEThe expressions involved contain a large number of terms, growing rapidly with \(n\). The present paper replaces Gao's ad hoc treatment, writing both the limit of the density function and the Katz-Sarnak prediction in the same combinatorial language. Most of the terms are shown to match for arbitrary \(n\), but for the remainder it is still necessary to perform explicit calculations. In this way it is established that the Katz-Sarnak conjecture holds for \(n\leq 7\), when the functions \(f_i\) are restricted as in \((*)\).