An Unexpected Connection Between the Discrete Zeta Function and the Erdos-Straus Conjecture Under Mballa's Conjecture

Abstract

In this article, we establish an additive decomposition of the discrete zeta function (for $s\in {\mathbb{N}}^{\ast }$, $s>1$), more precisely of the function $4(\hat{I}\P (s)-1)$, as a series whose general term is of the form $1/x_n(s)+1/y_n(s)+1/z_n(s)$, where $x_n(s),y_n(s),z_n(s)$ are solutions of the Erdos--Straus conjecture under a personal conjecture (which I will refer to here as Mballa's Conjecture) that I formulated by parametrization in the article: arXiv:2502.20935. This connection thus builds a bridge between analysis and Egyptian fractions in general, and the Erdos--Straus conjecture in particular.