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

TL;DR

This paper investigates the partial sums of the Dirichlet eta function series, their convergence properties, and their implications for the Riemann Hypothesis and the Simple Zeros Conjecture, providing new asymptotic and analytical insights.

Contribution

It introduces a novel analysis of the partial sums and remainders of the eta function series, linking their behavior to the Riemann Hypothesis and the Simple Zeros Conjecture.

Findings

Asymptotic relationship $R_n(s) o (-1)^{n-1}/n^s$ as $n o

Existence of the limit $L(s)$ related to the eta function ratios, connected to RH

Behavior of partial sums offers insights into the zeros of the zeta function

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 $n \rightarrow \infty$. 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.