{"paper":{"title":"On the Order of $a$ modulo $n$ on Average","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sungjin Kim","submitted_at":"2015-09-12T19:17:04Z","abstract_excerpt":"Let $a>1$ be an integer. Denote by $l_a(n)$ the multiplicative order of $a$ modulo integer $n\\geq 1$. We prove that there is a positive constant $\\delta$ such that if $x^{1-\\delta}\\log^3 x = o(y)$, then $$ \\frac1y \\sum_{a<y} \\frac1x \\sum_{\\substack{{a<n<x}\\\\{(a,n)=1}}}l_a(n) = \\frac x{\\log x}\\exp \\left(B\\frac{\\log\\log x}{\\log\\log\\log x}(1+o(1))\\right)$$ where $$ B=e^{-\\gamma}\\prod_p \\left(1-\\frac 1{(p-1)^2(p+1)}\\right).$$ This is an improvement over a statement in Kurlberg and Pomerance (see ~\\cite{KP}): $$\\frac{1}{x^2} \\sum_{a<x} \\sum_{a<n<x} l_a(n) = \\frac x{\\log x} \\exp \\left(B \\frac{\\log\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03768","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"}