Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family

Abstract

Let $q$ tend to infinity through odd primes, let $T=T(q)$, and put $\ell=\log(qT/2\pi)$. Suppose that $\ell^s=o(T)$ for some fixed $s>1$ and that \[ \lambda_{\rm bw}:=\min\left\{1, \liminf_{\substack{q\to\infty\\q\ \mathrm{prime}}} \frac{\log(q-1)}{\ell}\right\}>0. \] 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 $2-1/c_{\lambda_{\rm bw}}^*$ for both simple and distinct critical-line zeros and of $\tfrac12(3-1/c_{\lambda_{\rm bw}}^*)$ for all distinct zeros, where \[ c_\lambda^*=\frac{\sqrt2\tan(\lambda/\sqrt2)} {1+(\lambda/\sqrt2)\tan(\lambda/\sqrt2)}. \] 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 $X=\exp(\lambda\ell)$, 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.