Partial Sums of the Series for the Dirichlet Eta Function, their Peculiar Convergence, the Simple Zeros Conjecture, and the RH
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| Format: | Preprint |
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2009
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| _version_ | 1866914423122165760 |
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| author | Ghislanzoni, Luca |
| author_facet | Ghislanzoni, Luca |
| contents | For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $η_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $η(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by $u_n(s)$ the segment starting at $η_{n-1}(s)$ and ending at $η_n(s)$, 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 $χ_n^{\pm}(s) = η_n(1-s) / η_n(s)$. We then prove that the limit $L(s)=\lim_{N(s)<n\to\infty} χ_n^{\pm}(s)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_0907_2426 |
| institution | arXiv |
| publishDate | 2009 |
| record_format | arxiv |
| spellingShingle | Partial Sums of the Series for the Dirichlet Eta Function, their Peculiar Convergence, the Simple Zeros Conjecture, and the RH Ghislanzoni, Luca General Mathematics 11M06, 11M26 For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $η_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $η(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by $u_n(s)$ the segment starting at $η_{n-1}(s)$ and ending at $η_n(s)$, 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 $χ_n^{\pm}(s) = η_n(1-s) / η_n(s)$. We then prove that the limit $L(s)=\lim_{N(s)<n\to\infty} χ_n^{\pm}(s)$ 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. |
| title | Partial Sums of the Series for the Dirichlet Eta Function, their Peculiar Convergence, the Simple Zeros Conjecture, and the RH |
| topic | General Mathematics 11M06, 11M26 |
| url | https://arxiv.org/abs/0907.2426 |