Simple and distinct zeros in a prime-modulus Dirichlet family from near-microscopic to polylogarithmic heights

Abstract

Fix \(\eta>0\) and \(A_0>0\). Let \(q\) tend to infinity through odd primes, put \(Q=\log q\), and let \(T=T(q)\) satisfy \[ \frac{(\log Q)^{1+\eta}}{Q}\le T\le Q^{A_0}. \] Set \(I=(T,2T]\), and sum without weights over the \(q-2\) nonprincipal characters modulo \(q\). Let \(\mathcal N_q\) count nontrivial zeros in \(I\) with multiplicity, let \(\mathcal N^s_{0,q}\) and \(\mathcal N^*_{0,q}\) count simple and distinct zeros on the critical line, and let \(\mathcal N_{d,q}\) count all distinct zeros in \(I\). Uniformly in this height range, we prove unconditionally \[ \mathcal N_q=\frac{qT\log q}{2\pi}\{1+o_{\eta,A_0}(1)\}, \quad \frac{\mathcal N^s_{0,q}}{\mathcal N_q}, \frac{\mathcal N^*_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}}-o_{\eta,A_0}(1), \quad \frac{\mathcal N_{d,q}}{\mathcal N_q}\ge C_d-o_{\eta,A_0}(1), \] where \[ C_{\mathrm{MT}}=\frac32-\frac1{\sqrt2}\cot\!\left(\frac1{\sqrt2}\right) =0.672500703679\ldots, \qquad C_d=\frac{1+C_{\mathrm{MT}}}{2}=0.836250351839\ldots. \] The proof combines Selberg's family-averaged argument estimate and zero-density deletion with a finite Gevrey Gabor compression of Weil's Hermitian form. Quantitative Fourier--Laplace estimates control exterior zeros down to the lower endpoint, while a local--remote shell decomposition gives uniform control throughout the polylogarithmic upper range. Matrix moment estimates and an inertia-based rank--trace inequality then yield the counting bounds. No form of the generalized Riemann hypothesis is assumed.