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| Hovedforfatter: | |
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| Format: | Preprint |
| Udgivet: |
2020
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| Fag: | |
| Online adgang: | https://arxiv.org/abs/2002.04395 |
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Indholdsfortegnelse:
- 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.