Zeros of Koshliakov Zeta functions

Abstract

This thesis investigates the properties of zeros associated with the Riemann zeta function ζ(s) and the Hurwitz zeta function ζ(s, a). The thesis builds upon the work of N. S. Koshliakov [11, Chapter 1, 3], analyzing the properties of ζp(s) and ηp(s) alongside their analytic continuations with the relationship between ζp(s) and ηp(s). The research then delves into explicit formulas for the summation of specific infinite series. A key contribution of this thesis is the identification of a zero-free region for ζp(s) within the right half-plane.