Artin's Primitive Root Conjecture: Infinitude for Every Admissible Base

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Simultaneous primitive roots: a conditional lower bound for prime bases evidence for this event

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Artin's Primitive Root Conjecture: Infinitude for Every Admissible Base parent of this event

Primitive roots for every admissible integer base (companion supplying the Hecke zero-free input) evidence for this event

Deterministic factorization over prime fields, which uses this zero-free strip evidence for this event

VibeMathed record: Artin's Primitive Root Conjecture: Infinitude for Every Admissible Base evidence for this event

Theorem 1.1: for every integer $a$ that is neither $-1$ nor a square there are $c_a>0$ and $x_a$ such that for $x\ge x_a$ at least $c_a\,x/(\log x)^2$ primes $p\in(x,2x)$ have $a$ as a primitive root. This gives the infinitude part of Artin's conjecture for every admissible base, including $a=2$. It does not give the predicted asymptotic $A(a)\,x/\log x$: the count is short by a factor $\log x$ and constants depend on $a$. The analytic input is Theorem 1.2: for every cyclotomic field $F\supseteq\mu_{12}$ and every finite-order Hecke character $\eta$ of $F$, $L_F(s,\eta)$ has no zeros with $\mathrm{Re}\,s>1-10^{-6}$. The companion on simultaneous primitive roots gives the same lower bound for any finite set of prime bases, assuming four inputs from this paper. parent of this event