Potential Relation Between the Riemann Zeta Function and the Polynomial Function $F$ of the Generalized Erd\H{o}s--Straus Conjecture, Subject to its Analytic Continuation

Abstract

In this article, we explore a natural extension of the quadratic parametrization introduced in our previous work. By replacing the integer $n$ by $n^s$ ($ s\in\mathbb{R}, s>1$) and allowing the parameters to be real, we obtain for each $n\ge 1$ a decomposition $\frac{k}{n^s} = \frac{1}{x_s(n)}+\frac{1}{y_s(n)}+\frac{1}{z_s(n)}$ with $x_s(n), y_s(n), z_s(n) \in \mathbb{R}^*+$. Summing this equality over all integers brings forth the Riemann zeta function. Subject to an analytic continuation of the quantities $x_s(n), y_s(n), z_s(n)$ to complex values of $s$, one would obtain a new function \(G_k(s)\) satisfying $G_k(s)=k\,\zeta(s)$, thus establishing a deep connection between the structure of the conjecture and the zeros of $\zeta$.