{"paper":{"title":"Counting solutions without zeros or repetitions of a linear congruence and rarefaction in b-multiplicative sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandre Aksenov","submitted_at":"2014-03-03T19:46:59Z","abstract_excerpt":"Consider a strongly $b$-multiplicative sequence and a prime $p$. Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$. The integer values of the \"norm\" $3$-variate polynomial $\\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)\\!:=\\!\\prod_{j=1}^{p-1}\\left(Y_0{+}\\zeta_p^{i_1j}Y_1{+}\\zeta_p^{i_2j}Y_2\\right),$ where $\\zeta_p$ is a primitive $p$-th root of unity, and $i_1,i_2{\\in}\\{1,2,\\dots,p{-}1\\},$ determine this asymptotic behaviour. It will be shown that a combinatorial method can be applied to $\\mathcal N_{p,i_1,i_2}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0542","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"}