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Sarnak showed that the vector-valued function $E_{q;a_1,...,a_r}(x)=(E(x;q,a_1),..., E(x;q,a_r)),$ where $E(x;q,a)= \\frac{\\log x}{\\sqrt{x}}(\\phi(q)\\pi(x;q,a)-\\pi(x))$, has a limiting distribution $\\mu_{q;a_1,...,a_r}$ which is absolutely continuous on $\\mathbb{R}^r$. Under the same assumptions, we determine the asymptotic behavior of the large deviations $\\mu_{q;a_1,."},"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":"1103.0060","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-01T01:20:22Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"275dc6ec3612e79b3614dd0f39d2ac8a76022b6282225606819fa5615f51fadf","abstract_canon_sha256":"9b091468ab2f7c995cca975a86898022a9b913d5573e9268e1d6c91dfc5eda3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:37.936136Z","signature_b64":"N8oHPvIHX5teMNxtFp1hDtZtt3NJocoljYUXaED/ibBhY+9lIPDX4CyGeXAloh8lWAypqjAbn8nFDYNQARPBDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1483a8c8091edaef50e8b404b2e24347c5a1a479eb395909bbd64ce1e7ce2a7d","last_reissued_at":"2026-05-18T04:14:37.935652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:37.935652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large deviations of the limiting distribution in the Shanks-R\\'enyi prime number race","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Youness Lamzouri","submitted_at":"2011-03-01T01:20:22Z","abstract_excerpt":"Let $q\\geq 3$, $2\\leq r\\leq \\phi(q)$ and $a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime to $q$. Assuming the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis, M. Rubinstein and P. Sarnak showed that the vector-valued function $E_{q;a_1,...,a_r}(x)=(E(x;q,a_1),..., E(x;q,a_r)),$ where $E(x;q,a)= \\frac{\\log x}{\\sqrt{x}}(\\phi(q)\\pi(x;q,a)-\\pi(x))$, has a limiting distribution $\\mu_{q;a_1,...,a_r}$ which is absolutely continuous on $\\mathbb{R}^r$. Under the same assumptions, we determine the asymptotic behavior of the large deviations $\\mu_{q;a_1,."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0060","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":"1103.0060","created_at":"2026-05-18T04:14:37.935716+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.0060v2","created_at":"2026-05-18T04:14:37.935716+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.0060","created_at":"2026-05-18T04:14:37.935716+00:00"},{"alias_kind":"pith_short_12","alias_value":"CSB2RSAJD3NO","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"CSB2RSAJD3NO6UHI","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"CSB2RSAJ","created_at":"2026-05-18T12:26:26.731475+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/CSB2RSAJD3NO6UHIWQCLFYSDI7","json":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7.json","graph_json":"https://pith.science/api/pith-number/CSB2RSAJD3NO6UHIWQCLFYSDI7/graph.json","events_json":"https://pith.science/api/pith-number/CSB2RSAJD3NO6UHIWQCLFYSDI7/events.json","paper":"https://pith.science/paper/CSB2RSAJ"},"agent_actions":{"view_html":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7","download_json":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7.json","view_paper":"https://pith.science/paper/CSB2RSAJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.0060&json=true","fetch_graph":"https://pith.science/api/pith-number/CSB2RSAJD3NO6UHIWQCLFYSDI7/graph.json","fetch_events":"https://pith.science/api/pith-number/CSB2RSAJD3NO6UHIWQCLFYSDI7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7/action/storage_attestation","attest_author":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7/action/author_attestation","sign_citation":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7/action/citation_signature","submit_replication":"https://pith.science/pith/CSB2RSAJD3NO6UHIWQCLFYSDI7/action/replication_record"}},"created_at":"2026-05-18T04:14:37.935716+00:00","updated_at":"2026-05-18T04:14:37.935716+00:00"}