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On Viazovska's modular form inequalities.

Dan Romik1

  • 1Department of Mathematics, University of California, Davis, CA 95616.

Proceedings of the National Academy of Sciences of the United States of America
|October 18, 2023
PubMed
Summary

The Viazovska lattice sphere packing is proven to be the densest in 8 dimensions. This study provides a direct mathematical proof of key inequalities, avoiding computer calculations.

Keywords:
Eisenstein seriesJacobi thetanull functioninequalitymodular formsphere packing

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Area of Science:

  • Number Theory
  • Discrete Geometry
  • Mathematical Physics

Background:

  • The Viazovska lattice, a structure of points in 8-dimensional space, was previously shown to achieve the densest possible sphere packing.
  • This densest packing was proven using two critical inequalities involving modular and quasimodular forms.
  • The original proof relied on extensive computer-assisted calculations.

Purpose of the Study:

  • To provide a direct, human-verifiable proof of the inequalities central to the Viazovska sphere packing proof.
  • To establish the optimality of the Viazovska lattice sphere packing without computational aid.

Main Methods:

  • Development of direct analytical techniques for proving inequalities related to modular and quasimodular forms.
  • Focus on rigorous mathematical derivation rather than numerical computation.

Main Results:

  • A direct proof of the two key inequalities has been successfully established.
  • The proof confirms the Viazovska lattice sphere packing as the densest in 8 dimensions.

Conclusions:

  • The densest sphere packing in 8 dimensions is definitively confirmed through direct mathematical proof.
  • This work offers a more accessible and verifiable foundation for understanding this significant result in discrete geometry.