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Integral Representations and Asymptotics for a Family of Areal Mahler Measures

  • Quanyu Tang
  • Shu Zhang
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Abstract

We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family Pm(x1,…,xm,u) = ∏j=1m(1+xj) + u∏j=1m(1−xj), m≥1. Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for m𝔻(Pm). We prove that, for every fixed m, the value πmm𝔻(Pm) belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case m=2 explicitly in terms of Li4(1/2), ζ(3), Catalan's constant, π, and log2. We also derive an explicit three-term asymptotic expansion for m𝔻(Pm) as m→∞.

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Cite

BibTeXRIS
BibTeX
@misc{tang2026integral,
  title = {{Integral Representations and Asymptotics for a Family of Areal Mahler Measures}},
  author = {Quanyu Tang and Shu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.01951},
  eprint = {2608.01951},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.01951},
}
RIS
TY  - UNPB
TI  - Integral Representations and Asymptotics for a Family of Areal Mahler Measures
AU  - Quanyu Tang
AU  - Shu Zhang
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.01951
UR  - https://arxiv.org/abs/2608.01951
AB  - We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family P_m(x_1,…,x_m,u) = ∏_j=1^m(1+x_j) + u∏_j=1^m(1−x_j), m≥1. Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for m_𝔻(P_m). We prove that, for every fixed m, the value π^mm_𝔻(P_m) belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case m=2 explicitly in terms of Li_4(1/2), ζ(3), Catalan's constant, π, and log2. We also derive an explicit three-term asymptotic expansion for m_𝔻(P_m) as m→∞.
ER  -