Ordinality and Riemann Hypothesis II

For $\frac{1}{2}<x<1$, $y>0$, and $n\in \mathbb{N}$, let ${\displaystyle {\theta }_n(x+iy)=\sum _{i=1}^n\frac{sgn{q}_i}{q_i^{x+iy}}}$, where $Q=\{q_1,q_2,q_3,\cdots \}$ is the set of finite products of distinct odd primes, and $sgnq=(-1{)}^k$ if $q$ is the product of $k$ distinct primes. In this paper, we prove that there exists an ordering of $Q$ such that the sequence ${\theta }_n(x+iy)$ has a convergent subsequence. As an application, we study the Riemann hypothesis.