Irrationality of the dilogarithm at 1/3
Ruiyi Zhang · Preprint, 2026-10-08 · 24 pages
PDFHTMLDOI 10.5281/zenodo.23247754Source and checks
Abstract
We prove that Li_2(1/3) is irrational. Irrationality of Li_2(1/s) was known for integers s ≤ −5 and s ≥ 6 by work of Hata and of Rhin and Viola, and Prellberg recently proved it for s = −4, 5, −3. We define polynomials G_m(T) with rational coefficients by an interpolation determinant. The rank of this determinant modulo primes p with m/8 < p < 5m/4 bounds the denominators of the coefficients. A concavity inequality for a logarithmic energy, combined with the inversion v ↦ 3/v, bounds G_m(Li_2(1/3)). Interval arithmetic is used only for the energy and the potentials of a printed reference measure, on a uniform dyadic subdivision. The integral of the local rank function is a rational sum over 76 printed affine pieces. The coefficients of G_4 and the 9 × 9 determinant modulo 17 are printed and can be checked by hand. A congruence modulo 17 shows that G_m(ξ) ≠ 0 for every rational ξ and m = 4 · 17^t, t large in terms of ξ.
Cite
Ruiyi Zhang, Irrationality of the dilogarithm at 1/3, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23247754