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 $\Re(s)=α$. 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.

Figures (3)

• Figure 1: Pendulum model representing the gaps when $\alpha$ approaches $1$ on the right-hand side of the critical strip $\Re(s) \geq 1/2$.
• Figure 2: Roots of the biquadratic equation Q(x)=0. By the Riemann zeta functional, the roots of this biquadratic are the set of points equidistant to axis $\Re(s)=1/2$. The graph on the left-hand side shows a simple root, while the graph on the right two distinct roots. The case with two roots is characterized by distinct heights $\lambda_1$ and $\lambda_2$.
• Figure 3: Hypothetical scenario of a symmetrical curve about the vertical axis $\Re(s)=1/2$. As the roots of such biquadratics (extendable to polynomials of arbitrary orders) are given by equidistant points to the axis $\Re(s)=1/2$, the scenario of a symmetrical curve about $\Re(s)=1/2$ yields an infinity of solutions, i.e. any real $\alpha$ satisfies the biquadratic polynomial.

Theorems & Definitions (3)

• proof
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