Two thirds of the zeros of Dirichlet L-functions at polylogarithmic height are simple and on the critical line, on average

Abstract

Preprint (Version 2.3.1-draft, 18 August 2026). We outline a proof that for every sufficiently large prime q, at least 2/3 − O(1/log log q) of the zeros of the Dirichlet L-functions L(s, χ) in windows (T, 2T] at polylogarithmic height, averaged over the q−2 non-principal characters mod q, are simple and lie on the critical line, unconditionally — uniformly for every T with (log q)^(2+ε) ≤ T ≤ (log q)^A. This extends the family direction sketched in Remark 7.2(iii) of the August 10 version of the source manuscript (published as Alpöge–Furman, arXiv:2608.13637, whose Theorem B is the fixed-character t-aspect statement; the family-averaged polylog-window regime treated here is disjoint from it). Version 2.3.1 is a same-day restoration release, prompted by a dedicated presentation pass (every page of both PDFs rendered and inspected): LaTeX had silently dropped two footnotes placed inside the prior-work table — among them the footnote arguing why Alpöge–Furman's Theorem B does not reach this paper's asymptotic cell — so the markers printed but the texts were absent from v2.3. Both footnotes are restored, and the pass's other blocking findings are fixed (margin-overrunning file paths, stretched-glue page holes, glyph collisions in the prior-work table, one overfull display). New in the 2.3 line relative to the first deposit: the title states the predicate and the height regime ("low-lying" is reserved for the Katz–Sarnak one-level-density sense); a 219-word abstract with an extended summary; Appendix B.7, the full −(q−2)^(−1) bilinear-sums accounting with explicit constants; a T-range robustness audit extending the theorem from a fixed T to the full polylog range, with a common-window variant of the prime-conductor bridge corollary; Appendix D, a Montgomery–Taylor window variant reaching 0.6725007 and 0.8362503 with the Gevrey base constant shown profile-robust; and the minor findings of the latest full-document read applied. Produced with extensive AI assistance (Anthropic's Claude) under the direction of the author, with full transparency about the division of labor; see the manuscript's methodology section. 70 supporting lemmas are machine-checked in Lean 4. Internally reviewed through adversarial rounds documented in the deposited repository snapshot; not externally refereed. Files: the manuscript PDF (Version 2.3.1-draft), the review package (v6.1-sync), and a full repository snapshot (git commit fd639d5) containing the LaTeX sources, the Lean 4 development, the numerical experiments, and the complete audit trail.