A note on the zeros of generalized Hurwitz zeta functions

Given a function $f(n)$ periodic of period $q\ge 1$ and an irrational number $0<\alpha \le 1$, Chatterjee and Gun proved that the series $F(s,f,\alpha )={\sum }_{n=0}^{\mathrm{\infty }}\frac{f(n)}{(n+\alpha {)}^s}$ has infinitely many zeros for $\sigma >1$ when $\alpha$ is transcendental and $F(s,f,\alpha )$ has a pole at $s=1$, or when $\alpha$ is algebraic irrational and $c=\frac{\max f(n)}{\min f(n)}<1.15$. In this note, we prove that the result holds in full generality.