Partial Sums of the Series for the Dirichlet Eta Function, their Peculiar Convergence, the Simple Zeros Conjecture, and the RH

For any $s\in \mathbb{C}$ with $\mathrm{\Re }(s)>0$, denote by ${\eta }_{n-1}(s)$ the $(n-1{)}^{th}$ partial sum of the Dirichlet series for the eta function $\eta (s)=1-2^{-s}+3^{-s}-\cdots$, and by $R_n(s)$ the corresponding remainder. Denoting by $u_n(s)$ the segment starting at ${\eta }_{n-1}(s)$ and ending at ${\eta }_n(s)$, we first show how, for sufficiently large $n$ values, the circle of diameter $u_{n+2}(s)$ lies strictly inside the circle of diameter $u_n(s)$, to then derive the asymptotic relationship $R_n(s)\sim (-1{)}^{n-1}\mathrm{/}{n}^s$, as $n\to \mathrm{\infty }$. Denoting by $D=\{s\in \mathbb{C}:0<\mathrm{\Re }(s)<\frac{1}{2}\}$ the open left half of the critical strip, define for all $s\in D$ the ratio ${\chi }_n^{\pm }(s)={\eta }_n(1-s)\mathrm{/}{\eta }_n(s)$. We then prove that the limit $L(s)={\lim }_{N(s)<n\to \mathrm{\infty }}{\chi }_n^{\pm }(s)$ exists at every point $s$ of the domain $D$. The function $L(s)$ is continuous on $D$ if and only if the Riemann Hypothesis is true. Finally, we remark how the asymptotic behaviour of $R_n(s)$ can also provide insights substantiating the so called Simple Zeros Conjecture.