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 $\Lambda$.
Categories: math-ph math.MP
