Potential Relation Between the Riemann Zeta Function and the Polynomial Function $F$ of the Generalized Erdős--Straus Conjecture, Subject to its Analytic Continuation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914347650908160 |
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| author | Mballa, Philemon Urbain |
| author_facet | Mballa, Philemon Urbain |
| contents | 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 $G_k(s)=k\,ζ(s)$, thus establishing a deep connection between the structure of the conjecture and the zeros of $ζ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21112 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Potential Relation Between the Riemann Zeta Function and the Polynomial Function $F$ of the Generalized Erdős--Straus Conjecture, Subject to its Analytic Continuation Mballa, Philemon Urbain Number Theory 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 $G_k(s)=k\,ζ(s)$, thus establishing a deep connection between the structure of the conjecture and the zeros of $ζ$. |
| title | Potential Relation Between the Riemann Zeta Function and the Polynomial Function $F$ of the Generalized Erdős--Straus Conjecture, Subject to its Analytic Continuation |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.21112 |