{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:HJP3MQZB7KEBCOKAXJUZDBZB5J","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"507df65acd9fd334655269bebface5cf4586cc9d60ebfe9c007cca84e135733f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-09T18:16:13Z","title_canon_sha256":"7d660c596d2cbfae119a75a560826967c330ec7181ff17a9eaea0d87bedcd13e"},"schema_version":"1.0","source":{"id":"1701.02286","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.02286","created_at":"2026-05-18T00:53:11Z"},{"alias_kind":"arxiv_version","alias_value":"1701.02286v1","created_at":"2026-05-18T00:53:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.02286","created_at":"2026-05-18T00:53:11Z"},{"alias_kind":"pith_short_12","alias_value":"HJP3MQZB7KEB","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"HJP3MQZB7KEBCOKA","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"HJP3MQZB","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:534f7da738641577c0cdc276ca82b62ba3d06115c7bfbfe473df45ca9cc8d36d","target":"graph","created_at":"2026-05-18T00:53:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $q > 2$ be a prime number and define $\\lambda_q := \\left( \\frac{\\tau}{q} \\right)$ where $\\tau(n)$ is the number of divisors of $n$ and $\\left( \\frac{\\cdot}{q} \\right)$ is the Legendre symbol. When $\\tau(n)$ is a quadratic residue modulo $q$, then $\\left( \\lambda_q \\star \\mathbf{1} \\right) (n)$ could be close to the number of divisors of $n$. This is the aim of this work to compare the mean value of the function $\\lambda_q \\star \\mathbf{1}$ to the well known average order of $\\tau$. The proof reveals that the results depend heavily on the value of $\\left( \\frac{2}{q} \\right)$. A bound for s","authors_text":"Olivier Bordell\\`es","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-09T18:16:13Z","title":"When the number of divisors is a quadratic residue"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02286","kind":"arxiv","version":1},"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":"sha256:2174f5c6960c0227e45aa14e511a25ee9eab2f2e6e1e54f39b3e25fb94c676f1","target":"record","created_at":"2026-05-18T00:53:11Z","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":"507df65acd9fd334655269bebface5cf4586cc9d60ebfe9c007cca84e135733f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-09T18:16:13Z","title_canon_sha256":"7d660c596d2cbfae119a75a560826967c330ec7181ff17a9eaea0d87bedcd13e"},"schema_version":"1.0","source":{"id":"1701.02286","kind":"arxiv","version":1}},"canonical_sha256":"3a5fb64321fa88113940ba69918721ea5841a12dafb992c4fede7da832d6c886","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a5fb64321fa88113940ba69918721ea5841a12dafb992c4fede7da832d6c886","first_computed_at":"2026-05-18T00:53:11.308445Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:11.308445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SY7vHzuGF/TervApZHlE7fvP6V7ElTO2QOfaCbA6MZNliUiJaeaDfBpqRtI2a2hsY223LZ4tRT4WLVdiskVrBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:11.308867Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.02286","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2174f5c6960c0227e45aa14e511a25ee9eab2f2e6e1e54f39b3e25fb94c676f1","sha256:534f7da738641577c0cdc276ca82b62ba3d06115c7bfbfe473df45ca9cc8d36d"],"state_sha256":"3f6e0419c110af4805cf894bbc1df11786c021b811361807055eb9e013241811"}