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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}): $$\\frac{1}{x^2} \\sum_{a<x} \\sum_{a<n<x} l_a(n) = \\frac x{\\log x} \\exp \\left(B \\frac{\\log\\l"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.03768","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-12T19:17:04Z","cross_cats_sorted":[],"title_canon_sha256":"9eaf380f52f86559b19b5158a1ea06e4f5ff0939948cf537e3f040347bd43389","abstract_canon_sha256":"882d0e14814209d60e7633374ab50c8a8e3ccae80f474d8c6f0496352efdeb88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:29.285936Z","signature_b64":"EkA9DPNiDKMdJYvYGjrB4GPFZ3c80wLBW1OcnIiJSq5XQYCfCpggwWQVrJ0YmR7h/ZUZ7GyaLCIEKW5pVZ6FCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d6359ce96eeebed1b4e1ce3ab29f7c11ed067971dd4b983c884caca7ba97736","last_reissued_at":"2026-05-18T01:14:29.285370Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:29.285370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Order of $a$ modulo $n$ on Average","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sungjin Kim","submitted_at":"2015-09-12T19:17:04Z","abstract_excerpt":"Let $a>1$ be an integer. Denote by $l_a(n)$ the multiplicative order of $a$ modulo integer $n\\geq 1$. We prove that there is a positive constant $\\delta$ such that if $x^{1-\\delta}\\log^3 x = o(y)$, then $$ \\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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.03768","created_at":"2026-05-18T01:14:29.285440+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.03768v2","created_at":"2026-05-18T01:14:29.285440+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03768","created_at":"2026-05-18T01:14:29.285440+00:00"},{"alias_kind":"pith_short_12","alias_value":"NVRVTTUW53V6","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"NVRVTTUW53V62G2O","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"NVRVTTUW","created_at":"2026-05-18T12:29:34.919912+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE","json":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE.json","graph_json":"https://pith.science/api/pith-number/NVRVTTUW53V62G2ODTR2WKPXYE/graph.json","events_json":"https://pith.science/api/pith-number/NVRVTTUW53V62G2ODTR2WKPXYE/events.json","paper":"https://pith.science/paper/NVRVTTUW"},"agent_actions":{"view_html":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE","download_json":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE.json","view_paper":"https://pith.science/paper/NVRVTTUW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.03768&json=true","fetch_graph":"https://pith.science/api/pith-number/NVRVTTUW53V62G2ODTR2WKPXYE/graph.json","fetch_events":"https://pith.science/api/pith-number/NVRVTTUW53V62G2ODTR2WKPXYE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE/action/storage_attestation","attest_author":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE/action/author_attestation","sign_citation":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE/action/citation_signature","submit_replication":"https://pith.science/pith/NVRVTTUW53V62G2ODTR2WKPXYE/action/replication_record"}},"created_at":"2026-05-18T01:14:29.285440+00:00","updated_at":"2026-05-18T01:14:29.285440+00:00"}