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Peer-reviewedCC BY 4.0Forum of Mathematics, Pi

A variational principle for Gaussian lattice sums

  • Laurent Bétermin1
  • Markus Faulhuber2
  • Stefan Steinerberger3
  1. 1Université Claude Bernard Lyon 1
  2. 2University of Vienna
  3. 3University of Washington
Open at sourceDOICite

Abstract

We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices Λ ⊂ R 2 Λ ⊂ ℝ2 normal upper Lamda subset of double struck upper R squared with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be seen as the inhomogeneous counterpart to Montgomery’s 1988 result, which proved that the hexagonal lattice minimizes the maximum.

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Cite

BibTeXRIS
BibTeX
@article{betermin2026variational,
  title = {{A variational principle for Gaussian lattice sums}},
  author = {Laurent Bétermin and Markus Faulhuber and Stefan Steinerberger},
  journal = {Forum of Mathematics, Pi},
  year = {2026},
  month = oct,
  doi = {10.1017/fmp.2026.10040},
  url = {https://www.cambridge.org/core/product/identifier/S2050508626100407/type/journal_article},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - A variational principle for Gaussian lattice sums
AU  - Laurent Bétermin
AU  - Markus Faulhuber
AU  - Stefan Steinerberger
PY  - 2026
DA  - 2026/10/26
JO  - Forum of Mathematics, Pi
DO  - 10.1017/fmp.2026.10040
UR  - https://www.cambridge.org/core/product/identifier/S2050508626100407/type/journal_article
AB  - We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices Λ ⊂ R 2 Λ ⊂ ℝ^2 normal upper Lamda subset of double struck upper R squared with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be seen as the inhomogeneous counterpart to Montgomery’s 1988 result, which proved that the hexagonal lattice minimizes the maximum.
SN  - 2050-5086
ER  -