zeros of a complex function defined by integers

Question

Is there a known increasing sequence of positive integers $\{\textbf{a}\} = a_0<a_1<a_2<.....$ such that all the zeros $z_k$ on $\Re[z]>0$ of the complex function $F(z;\{\textbf{a}\})= \frac{1}{a_0^z}+\frac{1}{a_1^z}+\frac{1}{a_2^z}+.......$ with $F(z_k;\{\textbf{a}\})=0$ are such that $\Re[z_0]=\Re[z_1]=\Re[z_2]=... =c$ for some real number $c$? Also, if we are given an arbitrary real $c$ is there always a sequence $\{\textbf{a}\}$ with this property?

Answer 1

Yes, for example the sequence $a_n = 2^n$ gives the complex function $F(z) = \sum_{n \ge 0} 2^{-nz}$, which converges for $\Re(z) > 0$ to $F(z) = (1 - 2^{-z})^{-1}$, whose continuation on $\Bbb C \setminus \{2ik\pi/\log2 \mid k\in \Bbb Z\}$ doesn't have any zero at all, so those zeroes all are on all the lines $\Re(z)=c$, for every $c$ at once.