Quasi-Riemann Hypothesis

The quasi-Riemann hypothesis asserts that the Riemann zeta function has a fixed zero-free region of the form for some real number . It is a weakening of the Riemann hypothesis, which asserts that all Riemann zeta function zeros in the critical strip have real part and would therefore give (Davenport 1980, Billington et al. 2025).

A single constant must work at every height , unlike classical zero-free regions whose boundaries approach . The pole at is not a zero and is excluded when stating nonvanishing.

OpenAI (2026) announced nonvanishing in the half-plane

The claim covers the Riemann zeta function and all Dirichlet L-series, with the principal-character pole at excluded. It does not cover the boundary line or prove the Riemann hypothesis or the generalized Riemann hypothesis. The release attributes the argument and Lean formalization to an internal OpenAI model, an example of automated proof generation in AI-assisted mathematics. As of Oct. 25, 2026, independent specialist verification of the complete proof had not been reported.

In the plot above, the shaded area shows the announced zero-free region. The dashed critical line passes through the plotted Riemann zeta function zeros.

Terence Tao described the announced result as the biggest advance on the subject in 50 years. He characterized the argument as an unusual combination of existing ideas that he thought no single living mathematician would have conceived, while allowing that the right group of mathematicians might have found it after much trial and error. He also noted that the method could not move the zero-free boundary to the left of without another extremely clever idea (Numberphile 2026, 16:04-16:59).

The manuscript illustrates the combination Tao describes by drawing on analytic number theory, harmonic analysis, and complex analysis. It compares two expressions for a weighted sum built from the Fourier coefficients of Kubota's cubic theta function. The transformation of the function at its cusps and estimates from the large sieve, including a quadratic character estimate, bound the sum. A second expression, obtained by Poisson summation, isolates a Mellin transform involving the reciprocal of the target Hecke L-function. If the supremum of the real parts of the putative zeros exceeded the proposed bound, the estimates would give an analytic continuation of the reciprocal to a region extending a uniform distance to the left of that supremum, contradicting its definition. The manuscript first applies this argument at . It then modifies the sum at selected primes and uses bounds on moments of the resulting sums to reach (OpenAI 2026).

A stronger uniform version asks for the same for all Dirichlet L-series, independently of the number theoretic character, its modulus, and the height. The generalized Riemann hypothesis would give in this version as well.

The uniform version also has consequences for the size of primitive roots. Martin (1997) used sieve methods to bound the least prime number that is a primitive root modulo a prime number , assuming suitable zero-free regions for Dirichlet L-series. Since the least primitive root is no larger than the least prime primitive root, applying his bound conditionally to the announced uniform half-plane gives . The Vinogradov notation means that for some constant independent of . Since as for every , this implies , with the implied constant allowed to depend on . This is the bound asserted by Vinogradov's least primitive root conjecture.