Linnik's constant is at most $3.99$
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.…
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