Three Irrationality Results for the Dilogarithm: Rhin--Viola and Viola--Zudilin Constructions at −1/4, 1/5, and −1/3
Abstract
We prove the irrationality of Li2(−1/4), Li2(1/5), Li2(−1/3). The three arguments form a natural progression. The first is an endpoint completion of the five-parameter Rhin--Viola method. Rhin and Viola's 2019 treatment already supplies the negative-argument continuation, permutation invariance and factorial divisor; at z=−4 we use three bounded shifts, an elementary fixed-contour estimate and a four-term recurrence to remove the remaining nonvanishing problem in the complex-saddle regime. For 1/5 we pass to the six-parameter Viola--Zudilin family and apply the Rhin--Viola factorial transformations term by term inside two binomial expansions of the same integral. This inherited divisor crosses the arithmetic threshold that the unrefined five-parameter construction does not reach in our computations. The same crossbreed, combined with the negative-argument continuation, proves the result at −1/3. All proof-critical finite inequalities are accompanied by exact executable certificates. Supplementary computations propagate 10,000 exact primitive coefficient pairs for each of 1/5 and −1/3, with complete gcd removal and no failure of the certified smallness or adjacent nonproportionality checks.
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BibTeX
@misc{prellberg2026three,
title = {{Three Irrationality Results for the Dilogarithm: Rhin--Viola and Viola--Zudilin Constructions at $-1/4$, $1/5$, and $-1/3$}},
author = {Thomas Prellberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06895},
eprint = {2609.06895},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06895},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Three Irrationality Results for the Dilogarithm: Rhin--Viola and Viola--Zudilin Constructions at −1/4, 1/5, and −1/3
AU - Thomas Prellberg
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06895
UR - https://arxiv.org/abs/2609.06895
AB - We prove the irrationality of Li_2(−1/4), Li_2(1/5), Li_2(−1/3). The three arguments form a natural progression. The first is an endpoint completion of the five-parameter Rhin--Viola method. Rhin and Viola's 2019 treatment already supplies the negative-argument continuation, permutation invariance and factorial divisor; at z=−4 we use three bounded shifts, an elementary fixed-contour estimate and a four-term recurrence to remove the remaining nonvanishing problem in the complex-saddle regime. For 1/5 we pass to the six-parameter Viola--Zudilin family and apply the Rhin--Viola factorial transformations term by term inside two binomial expansions of the same integral. This inherited divisor crosses the arithmetic threshold that the unrefined five-parameter construction does not reach in our computations. The same crossbreed, combined with the negative-argument continuation, proves the result at −1/3. All proof-critical finite inequalities are accompanied by exact executable certificates. Supplementary computations propagate 10,000 exact primitive coefficient pairs for each of 1/5 and −1/3, with complete gcd removal and no failure of the certified smallness or adjacent nonproportionality checks.
ER -