Irrationality of the dilogarithm at 1/4
Ruiyi Zhang · Preprint, 2026-09-13 · 12 pages
PDFHTMLDOI 10.5281/zenodo.23247746Source and checks
Abstract
We prove that Li_2(1/4) 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. The linear forms in 1 and Li_2(1/4) come from one double integral of Viola and Zudilin at z = 4. We expand it into Rhin–Viola integrals in three ways and apply their factorial transformations to each term. This gives a common factor of the coefficients of the Nth form. The third expansion adds the primes between 265N and 322N to this factor, which is too small without them. We bound the integral by the weighted arithmetic-geometric mean inequality and by concavity on circular arcs.
Cite
Ruiyi Zhang, Irrationality of the dilogarithm at 1/4, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23247746