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

Abstract

For any $s\in \mathbb{C}$ with $\Re (s)>0$, 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 $R_n(s)\sim (-1{)}^{n-1}/n^s$, as $n\to \infty$. Denoting by $D=\left\{s\in \mathbb{C}:0<\Re (s)<\frac{1}{2}\right\}$ the open left half of the critical strip, define for all $s\in D$ the ratio ${\chi }_n^{\pm }(s)={\eta }_n(1-s)/{\eta }_n(s)$. We then prove that the limit $L(s)={\lim }_{N(s)<n\to \infty }{\chi }_n^{\pm }(s)$ 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.

12 pages. To further simplify the discussion in Final Remarks, additional elements have been added to the animation in Figure 5. The relevant narration has then been streamlined with help from Grammarly AI

References in corpus (1)

• Riemann Zeta Function