Next order energy asymptotics for Riesz potentials on flat tori

Let $\Lambda$ be a lattice in ${\bf R}^d$ with positive co-volume. Among $\Lambda$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $\mathcal{E}_{s,\Lambda}(N)$. While the dominant term in the asymptotic expansion of $\mathcal{E}_{s,\Lambda}(N)$ as $N$ goes to infinity in the long range case that $0<s<d$ (or $s=\log$) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for $s>0$ they are of the form $C_{s,d}|\Lambda|^{-s/d}N^{1+s/d}$ and $-\frac{2}{d}N\log N+\left(C_{\log,d}-2\zeta'_{\Lambda}(0)\right)N$ where we show that the constant $C_{s,d}$ is independent of the lattice $\Lambda$.