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

Abstract

The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines $\Re (s)=\alpha$. 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. $\Re (s)\in \mathcal{P}$, $|\eta (s)|\ge \left|1-\frac{\sqrt{2}}{2^{\alpha }}\right|$, where $\eta$ is the Dirichlet eta function, is related with the Riemann hypothesis as $|\eta (s)|>0$ 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. $\Re (s)\in ]1/2,1[$, $|\eta (s)|\ge \text{Min}\left(1-\frac{\sqrt{2}}{2^{\alpha }},\frac{\sqrt{2}}{2^{\alpha }}-\frac{\sqrt{2}}{2}\right)$, resulting from transitive composition in $\eta (s)=\left(1-\frac{2}{2^s}\right)\zeta (s)$. 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.

17 pages

References in corpus (1)

• An investigation of the non-trivial zeros of the Riemann zeta function