Irrationality of the trilogarithm at reciprocals of large integers
Ruiyi Zhang · Preprint, 2026-10-13 · 29 pages
PDFHTMLDOI 10.5281/zenodo.23329357Source and checks
Abstract
We prove that Li_3(1/a) is irrational for every integer a ≥ 67 and every integer a ≤ −70. Earlier ranges are |a| ≥ 1038 (Hata), a ≥ 1173 (Panzone), and a ≥ 486, a ≤ −471 (Marcovecchio). For 4 | m we form an m × m determinant. Its entries are linear forms in 1 and Li_3(1/a). The kernel has only triple poles, so no other polylogarithm enters. We bound a common denominator of the coefficients. The determinant does not vanish along a prime-power subsequence that depends on a. A logarithmic energy bounds its size. Exact density tables with directed rounding give the energy bounds at a = 67 and a = −70. A slope comparison extends them to all a ≥ 67 and all a ≤ −70.
Cite
Ruiyi Zhang, Irrationality of the trilogarithm at reciprocals of large integers, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23329357