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