On Viazovska's modular form inequalities

TL;DR

This paper provides a direct, computer-free proof of key inequalities involving modular forms used by Viazovska to establish the optimality of the $E_8$ lattice sphere packing in 8 dimensions.

Contribution

It offers a novel, explicit proof of inequalities critical to Viazovska's sphere packing result, avoiding computational methods.

Findings

Proved inequalities without computer assistance

Confirmed the validity of inequalities used in sphere packing proof

Enhanced understanding of modular form inequalities

Abstract

Viazovska proved that the $E_8$ lattice sphere packing is the densest sphere packing in 8 dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.