A Tight Reverse Minkowski Inequality for the Epstein Zeta Function

2022/01/13 by Eisenberg, Yael, Regev, Oded, Stephens-Davidowitz, Noah
#FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2201.05201

Abstract

We prove that if L ⊂ ℝⁿ is a lattice such that det(L') ≥ 1 for all sublattices L' ⊆ L, then ∑_\substacky\inL
y≠\mathbf0 (‖y‖²+q)^-s ≤ ∑_\substackz ∈ ℤⁿ
z≠0 (‖z‖²+q)^-s for all s > n/2 and all 0 ≤ q ≤ (2s-n)/(n+2), with equality if and only if L is isomorphic to ℤⁿ.

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