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

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.

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.