素数模Dirichlet族在介观收缩高度处的简单且不同的零点
Simple and distinct zeros in a prime-modulus Dirichlet family from near-microscopic to polylogarithmic heights
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中文总结 AI 辅助
该数学数论研究在不依赖广义黎曼假设的前提下，证明素数模Dirichlet族介观区间内非平凡零点的计数公式，给出简单零点、临界线零点及所有不同零点占比的下界。
中文摘要 AI 辅助
固定0<α<1，令q通过奇素数趋于无穷，记Q=log q，设区间I=(Q^(-α),2Q^(-α)]。对模q的q-2个非主特征不加权平均，令𝒩_q为I中非平凡零点的个数（计重数）；𝒩^s_{0,q}和𝒩^*_{0,q}分别计数简单且位于临界线上的零点，𝒩_{d,q}计数所有不同的零点（无位置限制）。我们无条件证明：𝒩_q = [q(log q)^(1-α)]/(2π) {1 + o_α(1)}；当q→∞时，lim inf 𝒩^s_{0,q}/𝒩_q ≥ C_MT，lim inf 𝒩^*_{0,q}/𝒩_q ≥ C_MT，lim inf 𝒩_{d,q}/𝒩_q ≥ C_d，其中C_MT=3/2 - (1/√2)cot(1/√2)=0.672500703679…，C_d=(1+C_MT)/2=0.836250351839…。物理高度趋于零，而I包含≍(log q)^(1-α)个局部平均间距。证明结合了小高度计数、零点密度删除与Weil埃尔米特型的有限光滑Gabor压缩，一阶、二阶矩阵矩与基于惯性的秩-迹不等式得到这三个计数界，未假设任何形式的广义黎曼假设。
英文摘要
Fix \(η>0\) and \(A_0>0\). Let \(q\) tend to infinity through odd primes, put \(Q=\log q\), and let \(T=T(q)\) satisfy \[ \frac{(\log Q)^{1+η}}{Q}\le T\le Q^{A_0}. \] 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 \[ \mathcal N_q=\frac{qT\log q}{2π}\{1+o_{η,A_0}(1)\}, \quad \frac{\mathcal N^s_{0,q}}{\mathcal N_q}, \frac{\mathcal N^*_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}}-o_{η,A_0}(1), \quad \frac{\mathcal N_{d,q}}{\mathcal N_q}\ge C_d-o_{η,A_0}(1), \] 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.
