Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence
Abstract
For any non-zero periodic function f : Z → C of period N ⩾ 1 and any non-zero algebraic number z_0 that doesn't lie on a half-line [e^2iℓπ/N , e^2iℓπ/N ∞[, 0 ⩽ ℓ ⩽ N -1, we give a lower bound of order sqrt{s/log(s)} on the dimension of the Q(z_0 )-vector space spanned by the numbers L(f, i, z_0 ) : sum_{m=1}^∞ f(m)z_0^m/m^i. This generalizes a result Fischler proved in 2026, corresponding to f identically equal to 1 and |z_0| ⩽ 1: in this case, the numbers L(f, i, z_0 ) = Li_i (z_0 ) are polylogarithm values. Except for a finite number of cuts in the complex plane, our result still holds when |z_0 | > 1, that is in a domain where the series definition above for the numbers L(f, i, z_0 ) doesn't converge anymore, and we make sense of these numbers through analytic continuation. To obtain this result, we construct linear combinations of the numbers L(f, i, z_0 ) using a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on an integral representation of the polylogarithm functions and on a "Shidlovskii lemma".
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BibTeX
@misc{mistiaen2026linear,
title = {{Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence}},
author = {Ludovic Mistiaen},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09774},
eprint = {2610.09774},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05766301v1},
note = {CC BY-ND 4.0},
}
RIS
TY - UNPB
TI - Linear independence of values of polylogarithms with periodic coefficients, outside the disk of convergence
AU - Ludovic Mistiaen
PY - 2026
DA - 2026/10/05
PB - HAL
DO - 10.48550/arxiv.2610.09774
UR - https://hal.science/hal-05766301v1
AB - For any non-zero periodic function f : Z → C of period N ⩾ 1 and any non-zero algebraic number z_0 that doesn't lie on a half-line [e^2iℓπ/N , e^2iℓπ/N ∞[, 0 ⩽ ℓ ⩽ N -1, we give a lower bound of order sqrt{s/log(s)} on the dimension of the Q(z_0 )-vector space spanned by the numbers L(f, i, z_0 ) : sum_{m=1}^∞ f(m)z_0^m/m^i. This generalizes a result Fischler proved in 2026, corresponding to f identically equal to 1 and |z_0| ⩽ 1: in this case, the numbers L(f, i, z_0 ) = Li_i (z_0 ) are polylogarithm values. Except for a finite number of cuts in the complex plane, our result still holds when |z_0 | > 1, that is in a domain where the series definition above for the numbers L(f, i, z_0 ) doesn't converge anymore, and we make sense of these numbers through analytic continuation. To obtain this result, we construct linear combinations of the numbers L(f, i, z_0 ) using a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on an integral representation of the polylogarithm functions and on a "Shidlovskii lemma".
ER -