On Landau's "local method" for obtaining zero-free regions

Abstract

We extend Landau's local analytic method for the deduction of zero-free region to a very general setting, and as a particular consequence, obtain a zero-free region and corresponding Prime Number Theorem for Beurling number systems with well-behaved integers. We also discuss the sharpness of the method and prove estimates for the local number of zeros, as well as a clustering result, which roughly put says that zeros close to the 1-line cannot be isolated but must occur in clusters.