{"paper":{"title":"Counting Divisors in the Outputs of a Binary Quadratic Form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander Walker, Chan Ieong Kuan, David Lowry-Duda, Tinghao Huang","submitted_at":"2023-10-20T16:31:06Z","abstract_excerpt":"For a fixed natural number $h$, we prove meromorphic continuation of the two-variable Dirichlet series $\\sum_m r_2(m) \\sigma_w(m + h) (m + h)^{-s + w}$ to $\\mathbb{C}^2$ and use this to obtain asymptotics for $\\sum_{m^2 + n^2 \\leq X} \\sigma_w(m^2 + n^2 + h)$. We approach this continuation through spectral theory. Our results are comparable to earlier work of Bykovskii, who used different methods to study the sums $\\sum_{n^2 \\leq X} \\sigma_w(n^2 + h)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.13632","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2310.13632/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"}