{"paper":{"title":"Smooth Sums over Smooth $k$-Free Numbers and Statistical Mechanics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Francesco Cellarosi","submitted_at":"2013-04-24T14:50:25Z","abstract_excerpt":"We provide an asymptotic estimate for certain sums over k-free integers with small prime factors. These sums depend upon a complex parameter \\alpha and involve a smooth cut-off f. They are a variation of several classical number-theoretical sums. One term in the asymptotics is an integral operator whose kernel is the \\alpha-convolution of the Dickman-de Bruijn distribution, and the other term is explicitly estimated. The trade-off between the value of \\alpha and the regularity of f is discussed. This work generalizes the results of tow previous papers by the author and Ya.G. Sinai, where k=2 a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.6610","kind":"arxiv","version":2},"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"}