A new arXiv preprint reports an exact mathematical result about a four-dimensional lattice called D4. After candidate lattices are normalized to covolume one, D4 has the smallest value of the theta function at every positive scale, and the smallest value of the Epstein zeta function when its parameter is greater than 2. Equality occurs only for lattices that are orthogonally equivalent to D4, meaning they differ only by a rigid change of coordinates.
The comparison covers every full-rank lattice in four-dimensional real space within that normalization, considered up to orthogonal transformations. It is a mathematical comparison across the complete class of admissible lattices, rather than an estimate drawn from a limited set of examples.
A complete four-dimensional comparison
The two functions are the paper’s central mathematical yardsticks. In plain terms, the study asks whether one normalized lattice can score lower than all of its four-dimensional alternatives under both measures. The theta result applies for every positive value of its scale parameter, while the Epstein zeta result applies when the zeta parameter is greater than 2.
The authors interpret the result as showing that the root lattice minimizes both functions among all lattices in dimension four. They also describe it as resolving the dimension-four Rankin-Sobolev problem, a broader claim about the mathematical question addressed by the paper.
How the comparison was built
To establish the result, the authors combine local and global comparisons with estimates based on tangent cones. They also use the concavity of the log determinant and determinant separation. These ingredients make up the proof framework used to compare D4 with the other lattices in the stated domain.
For the remaining stationary pairs, the argument uses a Gaussian moment bound to obtain strict comparison when the theta scale alpha is at least 5. It then projects along a shortest vector and applies a three-dimensional Fourier bound.
The Epstein zeta conclusion is linked to the theta result through Mellin integration of the theta difference. The paper says that this difference is strictly positive at every positive scale unless the lattice is orthogonally equivalent to D4, which supplies the equality case for the zeta comparison.
Beyond the two main functions
The analysis also extends the comparison to positive Gaussian mixtures under the stated finiteness condition. It states that equality in this broader comparison occurs precisely when the lattice is isometric to D4.
A result with a defined boundary
Part of the verification is computer-assisted. The appendix reports checks over specified domains, terminal inequalities, analytic error bounds, exact rational polynomial arithmetic and a separate interval-arithmetic implementation. The paper says an embedded archive contains the complete subdivisions, programs and execution records used for the verification.
The document header identifies arXiv:2609.37615v1, dated 29 September 2026, and the supplied publication record lists no journal. The work is presented as an arXiv preprint rather than a journal-listed publication.
The theorem’s scope is the stated one: full-rank, covolume-one lattices in dimension four, with the theta comparison covering positive scales and the zeta comparison covering parameters greater than 2. The result therefore addresses a defined mathematical class and parameter range.
The acknowledgements report partial support for S. Luo from NSFC grants 12261045 and 12001253 and the Jiangxi Jieqing Fund grant 20242BAB23001, and for J. Wei from the Hong Kong General Research Fund. They also disclose ChatGPT assistance with mathematical exposition, review of proof arguments and citations, and preparation of computational verification materials, including the appendices.
Paper data and sources
Original title: On minima of theta and Epstein zeta functions in dimension four
Authors: Senping Luo, Juncheng Wei
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
Original paper · Full text