Irrationality of the trilogarithm at reciprocals of integers of absolute value at least 27
Ruiyi Zhang · Preprint, 2026-10-15 · 33 pages
PDFHTMLDOI 10.5281/zenodo.23395794Source and checks
Abstract
We prove that Li_3(1/a) is irrational for every integer a ≥ 27 and every integer a ≤ −27. One determinant construction is used on both sides. The positive side takes m = 12Q and g = 4Q; the negative side takes m = 16Q and g = 5Q. We prove a conservative coefficientwise integerizer, nonvanishing along prime-power towers, and a logarithmic-energy estimate. On the negative side the estimate rests on a signed residue identity along a bent contour. Exact rational and directed interval certificates give four strict endpoint margins. Slope comparisons carry them along both rays.
Cite
Ruiyi Zhang, Irrationality of the trilogarithm at reciprocals of integers of absolute value at least 27, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23395794