Zero-free Regions for Fractional Hypergeometric Zeta Functions on the Left half of the Complex Plane

Abstract

Some zero-free regions were known on the right half of the complex plane in the form of vertical strips for fractional hypergeometric zeta functions. In this paper, we describe and demonstrate zero-free regions on the left half of the complex plane for fractional hypergeometric zeta functions. The fractional hypergeometric zeta function of order  has no zeros to the left half of the complex plane except the trivial zeros on the real axis.