Half-waves and spectral Riesz means on the 3-torus
For a full rank lattice $Λ\subset \mathbb{R}^d$ and $\mathbf{A} \in \mathbb{R}^d$, consider $N_{d,0;Λ,\mathbf{A}}(Σ) = \# ([Λ+\mathbf{A}] \cap Σ\mathbb{B}^d) = \# \{\mathbf{k}\in Λ: |\mathbf{k}+\mathbf{A}| \leq Σ\}$. Consider the iterated integrals \[ N_{d,k+1;Λ,\mathbf{A}}(Σ) = \int_0^ΣN_{d,k;Λ,\mathbf{A}}(σ) \,\mathrm{d} σ, \] for $k\in \mathbb{N}$. After an elementary derivation via the Poisson summation formula of the sharp large-$Σ$ asymptotics of $N_{3,k;Λ,\mathbf{A}}(Σ)$ for $k\geq 2$ (these having an $O(Σ)$ error term), we discuss how they are encoded in the structure of the Fourier transform $\mathcal{F}N_{3;Λ,\mathbf{A}}(τ)$. The analysis is related to Hörmander's analysis of spectral Riesz means, as the iterated integrals above are weighted spectral Riesz means for the simplest magnetic Schrödinger operator on the flat $3$-torus. That the $N_{3,k;Λ,\mathbf{A}}(Σ)$ obey an asymptotic expansion to $O(Σ^2)$ is a special case of a general result holding for all magnetic Schrödinger operators on all manifolds, and the subleading polynomial corrections can be identified in terms of the Laurent series of the half-wave trace at $τ=0$. The improvement to $O(Σ)$ for $k\geq 2$ follows from a bound on the growth rate of the half-wave trace at late times.