More Zero-Free Regions for Fractional Hypergeometric Zeta Functions

TL;DR

This paper extends the understanding of fractional hypergeometric zeta functions by identifying zero-free regions on the left half of the complex plane, complementing previous results on the right half.

Contribution

It introduces new zero-free regions on the left half-plane for fractional hypergeometric zeta functions, enhancing the theoretical understanding of their zero distribution.

Findings

Zero-free regions on the left half-plane established

No zeros except trivial zeros on the real axis

Complements previous right-half zero-free results

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 $a$ has no zeros to the left half of the complex plane except the trivial zeros on the real axis.