{"paper":{"title":"Infinite divisibility of Smith matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Shaofang Hong","submitted_at":"2008-08-26T16:16:31Z","abstract_excerpt":"Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A positive semi-definite matrix $A=(a_{ij})$ with $a_{ij}\\ge 0$ for all $i$ and $j$ is called infinitely divisible if the fractional Hadamard power $A^{\\circ r}=(a_{ij}^r)$ is positive semi-definite for every nonnegative real number $r$. Let $S=\\{x_1, ..., x_n\\}$ be a set of $n$ distinct positive integers. In this paper, we show that if $f$ is a multiplicativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.3550","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":""},"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"}