On the maximal zero-free half-planes of the Hurwitz zeta function

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2025年8月22日

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André Pierro de Camargo

摘要

We show that the function $a \longmapsto \alpha (a)$ $a ⟼ α (a)$ that characterizes the maximal open zero-free half-planes of the Hurwitz zeta function $\zeta (s,a)$ $ζ (s, a)$ is continuous at transcendental values of a and it is discontinuous at rational values of a. Some computable upper bounds for $\alpha (a)$ $α (a)$ when a is rational are also obtained.