A lower bound for the modulus of the Dirichlet eta function on partition $\mathcal{P}$ from 2-D principal component analysis

Abstract

The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines $\Re(s)=\alpha$. An approach based on a two-dimensional principal component analysis matching the dimensionality of the complex plane, which is built on a parametric ellipsoidal shape, has been undertaken to achieve this result. This lower bound, which is expressed as $\forall s \in \, \mathbb{C}$ s.t. $\Re(s) \in \mathcal{P}(s)$, $|\eta(s)| \geq |1- \frac{\sqrt{2}}{2^{\alpha}}|$, where $\eta$ is the Dirichlet eta function, has implications for the Riemann hypothesis as $|\eta(s)| >0$ for any such $s \in \mathcal{P}$, where $\mathcal{P}$ is a partition spanning one half of the critical strip on either sides of the critical line $\Re(s) = 1/2$ depending upon a variable delimiting regions, complementary by mirror symmetry with respect to $\Re(s) = 1/2$.Comment: 13 page