Uniform cubic bounds for irrational values of the dilogarithm
Abstract
We study explicit denominator conditions ensuring the irrationality of the dilogarithm at positive and negative rational points. We prove that Li2(p/q) is irrational whenever q>(4e4/27)p3, and obtain an explicit, slightly smaller threshold for Li2(−p/q). The positive argument specializes the Rhin--Viola integral family and uses a uniform estimate for a restricted prime divisor. The negative argument combines the same arithmetic construction with three adjacent integral moments and an explicit maximization. We also give finite-numerator refinements by interpolating certified partition bounds and by introducing a comparison measure with a quadratic tail. In particular, Li2(5/q) is irrational for every integer q≥418. The proof separates the established integral and determinant constructions from the parameter estimates and finite certificates used here.
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BibTeX
@misc{tan2026uniform,
title = {{Uniform cubic bounds for irrational values of the dilogarithm}},
author = {Hu Tan and Ying Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01066},
eprint = {2611.01066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01066},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform cubic bounds for irrational values of the dilogarithm
AU - Hu Tan
AU - Ying Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01066
UR - https://arxiv.org/abs/2611.01066
AB - We study explicit denominator conditions ensuring the irrationality of the dilogarithm at positive and negative rational points. We prove that Li_2(p/q) is irrational whenever q>(4e^4/27)p^3, and obtain an explicit, slightly smaller threshold for Li_2(−p/q). The positive argument specializes the Rhin--Viola integral family and uses a uniform estimate for a restricted prime divisor. The negative argument combines the same arithmetic construction with three adjacent integral moments and an explicit maximization. We also give finite-numerator refinements by interpolating certified partition bounds and by introducing a comparison measure with a quadratic tail. In particular, Li_2(5/q) is irrational for every integer q≥418. The proof separates the established integral and determinant constructions from the parameter estimates and finite certificates used here.
ER -