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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","authors_text":"Sungjin Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-12T19:17:04Z","title":"On the Order of $a$ modulo $n$ on Average"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03768","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1b1c059e7a029cdcf421ec9382f362c4437cd07652969b0d5e2173f182004329","target":"record","created_at":"2026-05-18T01:14:29Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"882d0e14814209d60e7633374ab50c8a8e3ccae80f474d8c6f0496352efdeb88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-12T19:17:04Z","title_canon_sha256":"9eaf380f52f86559b19b5158a1ea06e4f5ff0939948cf537e3f040347bd43389"},"schema_version":"1.0","source":{"id":"1509.03768","kind":"arxiv","version":2}},"canonical_sha256":"6d6359ce96eeebed1b4e1ce3ab29f7c11ed067971dd4b983c884caca7ba97736","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d6359ce96eeebed1b4e1ce3ab29f7c11ed067971dd4b983c884caca7ba97736","first_computed_at":"2026-05-18T01:14:29.285370Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:29.285370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EkA9DPNiDKMdJYvYGjrB4GPFZ3c80wLBW1OcnIiJSq5XQYCfCpggwWQVrJ0YmR7h/ZUZ7GyaLCIEKW5pVZ6FCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:29.285936Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.03768","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b1c059e7a029cdcf421ec9382f362c4437cd07652969b0d5e2173f182004329","sha256:6a71ca30b179d092657ade3c9af71e4baeb12fa3f2556fb131e461a2edc88e03"],"state_sha256":"332150ca0a49b900726e0d78163c871967a0ab1b5e7df00ce2db7ac1f8697934"}