Infinitude of the zeros of the Lerch zeta function on the half plane ℜ(s)>1

For $a\in (0,1]$, the zeros of the Hurwitz zeta function $\zeta (s,a):={\sum }_{n\ge 0}{(n+a)}^{-s}$ have interesting features. There are no zeros in the half plane $\Re \left(s\right)\ge 1+a$, whereas there are infinitely many zeros in the strip $1<\Re \left(s\right)<1+a$, provided $a\ne 1/2,1$. The existence of these infinitely many zeros was first proved by Davenport and Heilbronn for rational and transcendental values of a and then by Cassels for algebraic irrational values of a. In this article, we consider the analogous question for the zeros of the cognate Lerch zeta function ${\zeta }_z(s,a):={\sum }_{n\ge 0}{z}^n{(n+a)}^{-s}$, where z is a complex number of unit modulus. When z is a root of unity, the question can be answered using a theorem of Zaghloul, which is an extension of a work of Chatterjee and Gun. In the general case, we need to further extend the method of Chatterjee and Gun.