A lower bound for the modulus of the Dirichlet eta function on a partition $\mathcal{P}$ from 2-D principal component analysis and transitive composition
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2020
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916409805635584 |
|---|---|
| author | Heymann, Yuri |
| author_facet | Heymann, Yuri |
| contents | The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines $\Re(s)=α$. The approach employs concepts of two-dimensional principal component analysis built on a parametric ellipse, to match the dimensionality of the complex plane. The one-sided lower bound $\forall s \in \mathbb{C}$ s.t. $\Re(s) \in \mathcal{P}$, $| η(s) | \geq \left| 1 - \frac{\sqrt{2}}{2^α} \right|$, where $η$ is the Dirichlet eta function, is related with the Riemann hypothesis as $|η(s)| > 0$ for any $s \in \mathbb{C}$ s.t. $\Re(s) \in \mathcal{P}$, where $\mathcal{P}$ is a partition spanning one half of the critical strip depending upon a variable. We propose the composite lower bound $\forall s \in \, \mathbb{C}$ s.t. $\Re(s) \in \,]1/2,1[$, $|η(s)| \geq \text{Min}\left(1- \frac{\sqrt{2}}{2^α},\frac{\sqrt{2}}{2^α}-\frac{\sqrt{2}}{2}\right)$, resulting from transitive composition in $η(s) = \left(1-\frac{2}{2^s} \right) ζ(s)$. As a founding principle, the solution space of the set of solutions referring to such $\mathcal{L}^2$-problem is a representation of the space spanned by explanatory variables satisfying its algebraic form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_04395 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A lower bound for the modulus of the Dirichlet eta function on a partition $\mathcal{P}$ from 2-D principal component analysis and transitive composition Heymann, Yuri General Mathematics 11M26, 11Axx The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines $\Re(s)=α$. The approach employs concepts of two-dimensional principal component analysis built on a parametric ellipse, to match the dimensionality of the complex plane. The one-sided lower bound $\forall s \in \mathbb{C}$ s.t. $\Re(s) \in \mathcal{P}$, $| η(s) | \geq \left| 1 - \frac{\sqrt{2}}{2^α} \right|$, where $η$ is the Dirichlet eta function, is related with the Riemann hypothesis as $|η(s)| > 0$ for any $s \in \mathbb{C}$ s.t. $\Re(s) \in \mathcal{P}$, where $\mathcal{P}$ is a partition spanning one half of the critical strip depending upon a variable. We propose the composite lower bound $\forall s \in \, \mathbb{C}$ s.t. $\Re(s) \in \,]1/2,1[$, $|η(s)| \geq \text{Min}\left(1- \frac{\sqrt{2}}{2^α},\frac{\sqrt{2}}{2^α}-\frac{\sqrt{2}}{2}\right)$, resulting from transitive composition in $η(s) = \left(1-\frac{2}{2^s} \right) ζ(s)$. As a founding principle, the solution space of the set of solutions referring to such $\mathcal{L}^2$-problem is a representation of the space spanned by explanatory variables satisfying its algebraic form. |
| title | A lower bound for the modulus of the Dirichlet eta function on a partition $\mathcal{P}$ from 2-D principal component analysis and transitive composition |
| topic | General Mathematics 11M26, 11Axx |
| url | https://arxiv.org/abs/2002.04395 |