{"paper":{"title":"The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"D. P. Hardin, E. B. Saff, J. S. Brauchart","submitted_at":"2008-08-08T20:15:02Z","abstract_excerpt":"We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\\to \\infty$. For $s\\ge -2$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\\neq0$, where $r$ is the Euclidean distance between points.\n  By analytic continuation we deduce the expansion for all complex values of $s$.\n  The Riemann zeta function plays an essential role in this asymptotic expansion."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.1291","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}