TOPICS
Search

Quasi-Riemann Hypothesis

The quasi-Riemann hypothesis asserts that the Riemann zeta function has a fixed zero-free half-plane Re[s]>sigma_0 for some real number sigma_0<1. Thus one constant sigma_0 must work at every height Im[s], unlike classical zero-free regions whose boundaries approach Re[s]=1. The pole at s=1 is not a zero and is excluded when stating nonvanishing. The Riemann hypothesis implies this assertion with sigma_0=1/2, but the quasi-Riemann hypothesis does not require that value (Davenport 1980, Billington et al. 2025).

A stronger uniform version asks for the same sigma_0 for all Dirichlet L-series, independently of the number theoretic character, its modulus, and the height. The generalized Riemann hypothesis would give sigma_0=1/2 in this version as well.

See also

Dirichlet L-Series, Generalized Riemann Hypothesis, Landau-Siegel Zero, Riemann Hypothesis, Riemann Zeta Function, Zero-Free Region

Explore with Wolfram|Alpha

References

Billington, K.; Cheng, M.; Schettler, J.; and Suriajaya, A. I. "The Average Number of Goldbach Representations and Zero-Free Regions of the Riemann Zeta-Function." Int. J. Number Th. 21, 289-316, 2025. https://doi.org/10.1142/S1793042125500150.Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.

Cite this as:

Weisstein, Eric W. "Quasi-Riemann Hypothesis." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quasi-RiemannHypothesis.html

Subject classifications