Irrationality of the dilogarithm at −1/2
Ruiyi Zhang · Preprint, 2026-10-08 · 23 pages
PDFHTMLDOI 10.5281/zenodo.23247748Source and checks
Abstract
We prove that Li_2(−1/2) 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/5 < p < 5m/4 bounds the denominators of the coefficients. A concavity inequality for a logarithmic energy bounds G_m(Li_2(−1/2)). The only interval-arithmetic step is the potential certificate. The exact data in Table A.1, the coefficients of G_4, and the 9 × 9 determinant modulo 11 are printed and hand-checkable. A congruence modulo 11 shows that G_m has no rational zero for m = 4 · 11^t.
Cite
Ruiyi Zhang, Irrationality of the dilogarithm at −1/2, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23247748