Riesz energy constant C_{s,d} for A2 and D4 lattices

Determine whether the formula C_{s,d} = |Λ|^{s/d} ζ_Λ(s/2) holds for the hexagonal lattice A2 (d=2) and the checkerboard lattice D4 (d=4); that is, prove that the constant C_{s,d} in the large-N asymptotic expansion of minimal Riesz s-energy equals the appropriately normalized Epstein zeta function of these lattices.

Background

In dimensions d=1, 8, and 24, the constant C_{s,d} governing the leading term in the large-N asymptotics of minimal Riesz s-energy is known to be a multiple of the Epstein zeta function of an optimal lattice (Z, E8, and the Leech lattice), thanks to universal optimality results. The authors recall this and then point out analogous expectations for d=2 and d=4.

They explicitly state the conjecture that the same Epstein-zeta formula should hold for the hexagonal lattice A2 in d=2 and the checkerboard lattice D4 in d=4, which would pin down C_{s,d} in these dimensions.

References

It is conjectured that eq:C.s.d.for.d.EQ.1.8.24 also holds for the hexagonal lattice \boldsymbol{A}_2 ($d=2$) and the checkerboard lattice \boldsymbol{D}_4 ($d=4$).

— On the lower bounds for the spherical cap discrepancy  (2502.15984 - Bilyk et al., 21 Feb 2025) in Section 5.3 (Comparison with conjectured asymptotic behaviour), discussion around Equation (C_{s,d} definition)

The conjectured value of $\mathcal{C}$ is

— Efficient Searches for Low-Energy Structures in Clusters with Thousands of Particles: Application to the Thomson Problem  (2610.08122 - Amore et al., 6 Oct 2026) in Section 3, Numerical Results, discussion surrounding Fig. 2 (the asymptotic-energy plot)