A convergent subsequence of $\theta_n(x+iy)$ in a half strip

For $\frac{1}{2}0$ and $n\in\mathbb{N}$, let $\displaystyle\theta_n(x+iy)=\sum_{i=1}^n\frac{{\mbox{sgn}}\, q_i}{q_i^{x+iy}}$, where $Q=\{q_1,q_2,q_3,\cdots\}$ is the set of finite product of distinct odd primes and ${\mbox{sgn}}\, q=(-1)^k$ if $q$ is the product of $k$ distinct primes. In this paper we prove that there exists an ordering on $Q$ such that $\theta_n(x+iy)$ has a convergent subsequence.