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For any nonnegative integer $k$, Let $$ M(x,q,k)=\\displaystyle\\mathop {\\displaystyle\\mathop{\\sum{'}}_{a=1}^{q} \\displaystyle\\mathop{\\sum{'}}_{b\\leq xq}}_{\\mbox{$\\tiny\\begin{array}{c} 2|a+b+1\\\\ ab\\equiv1(\\bmod q)\\end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymp"},"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":"2104.00216","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-04-01T03:00:38Z","cross_cats_sorted":[],"title_canon_sha256":"e06802aec059987d642d562e0ec24ee2f33791833fe52624503299d70f8bd31a","abstract_canon_sha256":"1586ca3f650724acb440aee61bbce41e3e74605bf4501bb0c0171556180e0f0a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:28:25.046590Z","signature_b64":"ovraGRAmbMy0xRrywHCP0kxvMe1RpGa1KrHOXsxfwKl3JHsBnq4i9uwXILWk9NnmWp+I+IG0uRwM0vWvn8OoAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0480080d2380983e54d2401957ef8df3e26c5c219d15f6db6ea98ceb7c35e6b","last_reissued_at":"2026-07-05T02:28:25.046201Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:28:25.046201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the difference between a D. H. Lehmer number and its inverse over short interval","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Haodong Wang, Rong Ma, Yana Niu","submitted_at":"2021-04-01T03:00:38Z","abstract_excerpt":"Let $q>2$ be an odd integer. For each integer $x$ with $0<x<q$ and $(q,x)= 1$, we know that there exists one and only one $\\bar{x}$ with $0<\\bar{x}<q$ such that $x\\bar{x}\\equiv1(\\bmod q)$. A Lehmer number is defined to be any integer $a$ with $2\\dagger(a+\\bar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=\\displaystyle\\mathop {\\displaystyle\\mathop{\\sum{'}}_{a=1}^{q} \\displaystyle\\mathop{\\sum{'}}_{b\\leq xq}}_{\\mbox{$\\tiny\\begin{array}{c} 2|a+b+1\\\\ ab\\equiv1(\\bmod q)\\end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.00216","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2104.00216/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2104.00216","created_at":"2026-07-05T02:28:25.046259+00:00"},{"alias_kind":"arxiv_version","alias_value":"2104.00216v1","created_at":"2026-07-05T02:28:25.046259+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.00216","created_at":"2026-07-05T02:28:25.046259+00:00"},{"alias_kind":"pith_short_12","alias_value":"UBEABAGSHAEY","created_at":"2026-07-05T02:28:25.046259+00:00"},{"alias_kind":"pith_short_16","alias_value":"UBEABAGSHAEYHZKN","created_at":"2026-07-05T02:28:25.046259+00:00"},{"alias_kind":"pith_short_8","alias_value":"UBEABAGS","created_at":"2026-07-05T02:28:25.046259+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/UBEABAGSHAEYHZKNEQAZK7XY34","json":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34.json","graph_json":"https://pith.science/api/pith-number/UBEABAGSHAEYHZKNEQAZK7XY34/graph.json","events_json":"https://pith.science/api/pith-number/UBEABAGSHAEYHZKNEQAZK7XY34/events.json","paper":"https://pith.science/paper/UBEABAGS"},"agent_actions":{"view_html":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34","download_json":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34.json","view_paper":"https://pith.science/paper/UBEABAGS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2104.00216&json=true","fetch_graph":"https://pith.science/api/pith-number/UBEABAGSHAEYHZKNEQAZK7XY34/graph.json","fetch_events":"https://pith.science/api/pith-number/UBEABAGSHAEYHZKNEQAZK7XY34/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34/action/storage_attestation","attest_author":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34/action/author_attestation","sign_citation":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34/action/citation_signature","submit_replication":"https://pith.science/pith/UBEABAGSHAEYHZKNEQAZK7XY34/action/replication_record"}},"created_at":"2026-07-05T02:28:25.046259+00:00","updated_at":"2026-07-05T02:28:25.046259+00:00"}