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

TL;DR

This paper reveals a novel link between the discrete zeta function and the Erdos-Straus conjecture by expressing a modified zeta function as a series involving solutions to the conjecture under a new conjecture, Mballa's Conjecture.

Contribution

It introduces a new additive decomposition of the discrete zeta function related to Egyptian fractions and connects it to the Erdos--Straus conjecture through a new conjecture, Mballa's Conjecture.

Findings

Established an additive decomposition of the discrete zeta function.

Linked the zeta function to solutions of the Erdos--Straus conjecture.

Proposed Mballa's Conjecture as a key component in this connection.

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 $4(\zeta(s)-1)$, 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.