{"paper":{"title":"Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Infinite lattice sums for dipolar and power-law interactions can be computed exactly using a small direct sum plus zeta function derivative corrections.","cross_cats":["cond-mat.str-el","cs.NA"],"primary_cat":"math.NA","authors_text":"Andreas A. Buchheit, Filipp N. Rybakov, Jonathan K. Busse, Torsten Ke{\\ss}ler","submitted_at":"2025-09-30T13:58:28Z","abstract_excerpt":"We address the efficient computation of power-law-based interaction potentials of homogeneous $d$-dimensional bodies with an infinite $n$-dimensional array of copies, including their higher-order derivatives. This problem forms a serious challenge in micromagnetics with periodic boundary conditions and related fields. Nowadays, it is common practice to truncate the associated infinite lattice sum to a finite number of images, introducing uncontrolled errors. We show that, for general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law po"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the required derivatives of generalized zeta functions (and related special functions such as incomplete Bessel functions) admit a superexponentially convergent algorithm that remains practical and stable for the geometries and derivative orders needed in micromagnetics.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Zeta-function derivative corrections enable machine-precision evaluation of periodic dipolar and Riesz potentials at the cost of truncated sums.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Infinite lattice sums for dipolar and power-law interactions can be computed exactly using a small direct sum plus zeta function derivative corrections.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"238704d15e62520cece48faa75419958e4c70c34a71d9cfc34a45c59e5e6ef5f"},"source":{"id":"2509.26274","kind":"arxiv","version":3},"verdict":{"id":"450a8d96-9bfa-4ae6-8fd9-bf4055ab5cb1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T12:07:34.710621Z","strongest_claim":"For general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions.","one_line_summary":"Zeta-function derivative corrections enable machine-precision evaluation of periodic dipolar and Riesz potentials at the cost of truncated sums.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the required derivatives of generalized zeta functions (and related special functions such as incomplete Bessel functions) admit a superexponentially convergent algorithm that remains practical and stable for the geometries and derivative orders needed in micromagnetics.","pith_extraction_headline":"Infinite lattice sums for dipolar and power-law interactions can be computed exactly using a small direct sum plus zeta function derivative corrections."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.26274/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}