Zero-Free Region

A zero-free region for a complex function is a subset of its domain on which . For the Riemann zeta function and Dirichlet L-series, such regions near the line provide estimates for the distribution of prime numbers.

A classical zero-free region for the Riemann zeta function is

where is an absolute constant and the pole at is excluded. This region narrows toward as grows (Davenport 1980). A fixed zero-free half-plane with is the assertion of the quasi-Riemann hypothesis. For Dirichlet L-series, the possible Landau-Siegel zero is an exception to the corresponding classical region.