Half-waves and spectral Riesz means on the 3-torus

For a full rank lattice Λ ⊂R^d and A ∈R^d , consider N_{d ,0 ;Λ ,A}(Σ ) =#([Λ +A ] ∩Σ B^d) =#{k ∈Λ :|k +A |≤Σ } . Consider the iterated integrals N_{d ,k +1 ;Λ ,A}(Σ ) =∫0_ΣN_{d ,k ;Λ ,A}(σ ) dσ , for k ∈N . After an elementary derivation via the Poisson summation formula of the sharp large-Σ asymptotics of N_{3 ,k ;Λ ,A}(Σ ) for k ≥2 (these having an O (Σ ) error term), we discuss how they are encoded in the structure of the Fourier transform F N_{3 ,k ;Λ ,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 d-torus. That the N_{3 ,k ;Λ ,A}(Σ ) obey an asymptotic expansion to O (Σ²) 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 ≥2 follows from a bound on the growth rate of the half-wave trace at late times.