A Note on a Recent Attempt to Prove the Irrationality of $ζ(5)$
Keyu Chen, Wei He, Yixin He, Yuxiang Huang, Yanyang Li, Quanyu Tang, Lei Wu, Shenhao Xu, Shuo Yang, Zijun Yu
TL;DR
This note critically examines a claimed proof of the irrationality of $\zeta(5)$, pinpointing a fundamental logical gap in conflating the solvability of a Diophantine equation with the irrationality of a zeta value. It reviews a standard irrationality criterion based on rapidly converging rational approximations and explains why the proposed approach fails, leaving the open problem of $\zeta(5)$ unresolved. The article also surveys contemporary advances by Calegari, Dimitrov, and Tang, which introduce powerful methods for proving irrationality and $\mathbb{Q}$-linear independence of zeta and $L$-values, offering a promising framework for future breakthroughs. Together, the work clarifies the limitations of the Suman approach and points toward arithmetic holonomy-bound techniques as a fruitful path forward.
Abstract
Recently Shekhar Suman [arXiv: 2407.07121v6 [math.GM] 3 Aug 2024] made an attempt to prove the irrationality of $ζ(5)$. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.