{"id":"hexagon/2610.00018","pdf":"https://hexagonmath.org/pdf/2610.00018v1","url":"https://hexagonmath.org/abs/2610.00018v1","tags":["msc:11N13","msc:11M26","msc:11M20","msc:11N35","msc:11Y35"],"score":0,"title":"Linnik's constant is at most $3.99$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Eric Naslund"],"reviews":0,"subject":"math.NT","summary":"Let $P(a,q)$ denote the least prime congruent to $a$ modulo $q$, where $(a,q)=1$. We give a computer-assisted proof that $P(a,q)\\ll q^{3.99}$, improving the exponent $5$ of Xylouris. More precisely, $P(a,q)<q^{3.99}$ for every sufficiently large $q$, uniformly in $a$. The main new ingredient is a graded near-density estimate for zeros of Dirichlet $L$-functions, a weighted form of Heath-Brown's Lemma~12.1 of the kind he asked for in 1992. We combine it with the zero-location estimates and far-density method of Heath-Brown and Xylouris. The second novelty is the scale of the case analysis.…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","ChatGPT","Grok"],"discussed":0,"published":"2026-10-02","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-01-000020&version=1"],"human_understanding":"some"}