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

For a full rank lattice Lambda subset of R-d and A is an element of R-d, consider Nd-,Nd- 0; Lambda,Nd-A(Sigma) = #([Lambda + A] n boolean AND Sigma B-d) = #{k is an element of Lambda : |k + A| = <= Sigma}. Consider the iterated integrals N-d,N- k+1; Lambda,A(Sigma) = integral(Sigma)(0)(Nd, k; Lambda,A)(sigma) d sigma, for k is an element of N. After an elementary derivation via the Poisson summation formula of the sharp large-Sigma asymptotics of N-3,N- k; Lambda,N-A(Sigma) for k >= 2 (these having an O(Sigma) error term), we discuss how they are encoded in the structure of the Fourier transform F-N3,F- k; Lambda,F-A(tau). The analysis is related to Hormander's analysis of spectral Riesz means, as the iterated integrals above are weighted spectral Riesz means for the simplest magnetic Schrodinger operator on the flat d-torus. That the N-3,N- k; Lambda,N-A(Sigma) obey an asymptotic expansion to O(Sigma(2)) is a special case of a general result holding for all magnetic Schrodinger operators on all manifolds, and the subleading polynomial corrections can be identified in terms of the Laurent series of the half-wave trace at t = 0. The improvement to O(Sigma) for k = 2 follows from a bound on the growth rate of the half-wave trace at late times.