Gevrey 局部化：素数模 Dirichlet 族中简单与不同零点的定位
Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family
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中文总结 AI 辅助
本文在素数模 Dirichlet 特征族中，通过 Gevrey 衰减与采样密度等新技术，证明了临界线上简单与不同零点的无条件下界，并给出了数值常数。
中文摘要 AI 辅助
令 $q$ 通过奇素数趋于无穷，令 $T=T(q)$，并设 $\ell=\log(qT/2\pi)$。假设对某个固定的 $s>1$ 有 $\ell^s=o(T)$，且 \\[ \lambda_{\rm bw}:=\min\left\{1, \liminf_{\substack{q\to\infty\q\\ \mathrm{prime}}} \frac{\log(q-1)}{\ell}\right\}>0. \\] 对于模 $q$ 的 $q-2$ 个非主特征的未加权族，我们证明了相对于 $(T,2T]$ 中总零点重数的无条件下界：对于简单临界线零点和不同临界线零点，下界为 $2-1/c_{\lambda_{\rm bw}}^*$；对于所有不同零点，下界为 $\tfrac12(3-1/c_{\lambda_{\rm bw}}^*)$，其中 \\[ c_\lambda^*=\frac{\sqrt2\tan(\lambda/\sqrt2)} {1+(\lambda/\sqrt2)\tan(\lambda/\sqrt2)}. \\] 当 $\log T=o(\log q)$ 时，这分别给出 $0.6725007036\ldots$ 和 $0.8362503518\ldots$；特别地，它覆盖了每个固定的 $T=(\log q)^A$（其中 $A>1$）。\n\n作为作者公开的介观收缩高度定理的伴随结果（针对增长高度情形），本文围绕 $T$ 增长时所需的不同定位技术进行组织。一个抽象定理将复带 Gevrey 衰减、采样密度和局部多重集计数转化为迹范数和核范数中的拉伸指数定位。在 Dirichlet 特化中，带增长为 $X^{1/4}$，其中 $X=\exp(\lambda\ell)$，缓冲区 $D_0=(K\ell)^s$ 均匀地吸收特征中的远程零点贡献。定量的有限采样端效应界、有限显式公式矩阵、特征平均的第一和第二迹、双曲函数方程块以及秩-迹不等式随后产生这三个零点统计量。共享的有限惯性机制和 Montgomery–Taylor 常数不被视为新结果。不假设任何形式的 GRH。
英文摘要
Let $q$ tend to infinity through odd primes, let $T=T(q)$, and put $\ell=\log(qT/2π)$. Suppose that $\ell^s=o(T)$ for some fixed $s>1$ and that \[ λ_{\rm bw}:=\min\left\{1, \liminf_{\substack{q\to\infty\\q\ \mathrm{prime}}} \frac{\log(q-1)}{\ell}\right\}>0. \] For the unweighted family of the $q-2$ nonprincipal characters modulo $q$, we prove unconditional lower bounds, relative to the total zero multiplicity in $(T,2T]$, of $2-1/c_{λ_{\rm bw}}^*$ for both simple and distinct critical-line zeros and of $\tfrac12(3-1/c_{λ_{\rm bw}}^*)$ for all distinct zeros, where \[ c_λ^*=\frac{\sqrt2\tan(λ/\sqrt2)} {1+(λ/\sqrt2)\tan(λ/\sqrt2)}. \] When $\log T=o(\log q)$, this gives respectively $0.6725007036\ldots$ and $0.8362503518\ldots$; in particular it covers every fixed $T=(\log q)^A$ with $A>1$. As the growing-height companion to the authors' public mesoscopic shrinking-height theorem, this article is organized around the different localization technology required when $T$ grows. An abstract theorem converts complex-strip Gevrey decay, sampling density, and a local multiset count into stretched-exponential localization in trace and nuclear norm. In the Dirichlet specialization the strip growth is $X^{1/4}$, where $X=\exp(λ\ell)$, and a buffer $D_0=(K\ell)^s$ absorbs the remote-zero contribution uniformly in the character. Quantitative finite-sampling end-effect bounds, a finite explicit-formula matrix, character-averaged first and second traces, hyperbolic functional-equation blocks, and a rank--trace inequality then yield the three zero statistics. The shared finite-inertia mechanism and the Montgomery--Taylor constant are not claimed as new. No form of GRH is assumed.
发表机构
- Changkong College, Nanjing University of Aeronautics and Astronautics（南京航空航天大学长空学院）
- School of Science, Nanjing University of Posts and Telecommunications（南京邮电大学理学院）
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