A classic mathematical constant used to calculate physical forces has been proven to be irrational using a new, higher-dimensional geometric proof.
April 24, 2026
Original Paper
Mellin transforms, transfinite diameter and rational approximations of integrals
arXiv · 2604.20741
The Takeaway
Higher-dimensional geometric criteria provide a new way to prove the irrationality of zeta(2). This constant is a cornerstone of physics and number theory, but its fundamental nature was traditionally proven using one-dimensional calculus. This new method uses transfinite diameters and Mellin transforms to look at the number through the lens of multi-dimensional shapes. It reveals a deep connection between the irregularity of numbers and the geometry of space. This framework could eventually be used to solve the much harder problem of proving the irrationality of other constants like zeta(3).
From the abstract
We establish a higher-dimensional irrationality criterion for periods which are presented as Mellin integrals depending on many parameters. The criterion is stated as an upper bound on the multi-variate transfinite diameter of the image of the domain of integration under the Mellin arguments. Most of the paper is devoted to studying notions of transfinite diameter relative to very general multivariate Vandermonde matrices.As a proof of principle, we illustrate how this approach works with detail