Vinogradov's Least Primitive Root Conjecture

Vinogradov's least primitive root conjecture asserts that the least positive primitive root modulo a prime number satisfies

(1)

for every . The implied constant may depend on but not on . This is stronger than the corresponding conjecture for the least quadratic nonresidue, since a primitive root modulo an odd prime is a quadratic nonresidue. Burgess (1962) proved the unconditional bound .

A uniform zero-free half-plane for Dirichlet L-series implies the conjecture. Specifically, suppose that some fixed satisfies for every nonprincipal number theoretic character and every , independently of the modulus and height. Martin's bound for the least prime primitive root (Martin 1997, p. 278) then gives

(2)

Indeed, for odd , write for the number of distinct prime factors and , where is the totient function. The expression inside Martin's exponent is . The estimates and bound this expression by a constant times .

OpenAI (2026) claimed the required uniform zero-free half-plane with in its proposed proof of the quasi-Riemann hypothesis. Conditional on that claim, the preceding argument yields

(3)

The release attributes the zero-free theorem and its Lean formalization to an internal OpenAI model. As of Oct. 25, 2026, independent specialist verification of that underlying theorem had not been reported. The implication therefore does not supply an independent unconditional proof of Vinogradov's conjecture.