math.NT — Number Theory

Every nonsquare integer not equal to $-1$ is a primitive root modulo a positive lower density of primes

Contributed by Manjul Bhargava

For every integer $a$ that is neither a square nor $-1$, we prove that a positive lower density of primes have $a$ as a primitive root. More precisely, a zero-free half-plane $\Re s>1-1/N$ for all Dedekind zeta functions gives a lower proportion of at least $c_a/(N\log N)$, with $c_a>0$ independent of $N\ge2$. The theorem of Cureton and Thorner supplies $N=6000$, so the positive-proportion conclusion is unconditional. The proof uses the marked Type II estimates of OpenAI to count primes $p$ such that $p-1$ is a small cofactor times one large prime. Allowing the cofactor to vary supplies an additional factor of $\log x$ in the count. We give the counting and sieve deductions in detail and explain the dependence on the zero-free width. The argument gives a lower bound rather than the exact Artin density.