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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 $-\\f"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1511.01552","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-11-04T23:30:20Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"23b7403c42bff906840e34fbc393fd4a56ce61c3af79da23c19e5677d91dff82","abstract_canon_sha256":"9ac97dae7bdcd10eb19b370d16fe167159778d0d64bc2784c789a3f860914959"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:45.299631Z","signature_b64":"VytnTRLeutlzosIbH1uiLMHeeM5vlru2TmAsyycy1EJxoGwrFt2kQGeD18zEjUrDDY8AStACUokintp4a9HTCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26fe15c129b59c113f400726bc6947fe571e0fb1537d14db3da8e98a2ceb7d8c","last_reissued_at":"2026-05-18T01:27:45.298990Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:45.298990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Next order energy asymptotics for Riesz potentials on flat tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Brian Z. 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