Ordinality and Riemann Hypothesis II

Abstract

For $\frac{1}{2}<x<1$, $y>0$, 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 products 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 of $Q$ such that the sequence $\theta_n(x+iy)$ has a convergent subsequence. As an application, we study the Riemann hypothesis.