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

TL;DR

The paper embeds the Erdős–Straus conjecture within a discrete zeta framework by leveraging Mballa's conjecture to parameterize $\frac{4}{n^s}$ as a sum of three unit fractions. It defines a discrete zeta function $\zeta_M(s)$ from this decomposition and proves the key identity $\zeta_M(s) = 4(\zeta(s) - 1)$, establishing a bridge between Egyptian fractions and the classical zeta function for $s>1$. A notable consequence is that $4(\zeta(2k+1) - 1) = \zeta_M(2k+1)$ for odd integer arguments, complemented by numerical evidence and an accessible verification appendix. The approach offers a new analytic lens on the Erdős–Straus conjecture and on odd zeta values, highlighting connections between Diophantine decompositions and zeta-analytic structures.

Abstract

In this article, we establish an additive decomposition of the discrete zeta function (for $s \in \mathbb{N}^*$, $s > 1$), more precisely of the function $4(ζ(s)-1)$, 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.