A Modular Form Proof of the Irrationality of $ζ\left(3\right)$

Pang Ern Thang

A Modular Form Proof of the Irrationality of $ζ\left(3\right)$

Number Theory 2026-07-19 v1

Authors: Pang Ern Thang

Abstract

We present an expository proof of the irrationality of $\zeta\left(3\right)$ using modular forms of level 6. By constructing a suitable Eichler integral, we obtain a power series with controlled denominators and sufficiently large radius of convergence. Beukers' irrationality criterion then implies that $\zeta\left(3\right)$ is irrational.

Keywords

irrationality zeta(3) modular forms Eichler integral Beukers criterion

Cite

@article{arxiv.2607.17123,
  title  = {A Modular Form Proof of the Irrationality of $ζ\left(3\right)$},
  author = {Pang Ern Thang},
  journal= {arXiv preprint arXiv:2607.17123},
  year   = {2026}
}