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We prove Weak Weyl's Law for $N_{\\mathrm{sd}}^K(\\lambda)$ in the form that there are positive constants $c_1, c_2$ (depending on $K$) and $d$ such that $c_1\\lambda^{d/2}\\leq N_{\\mathrm{sd}}^K(\\lambda)\\leq c_2\\lambda^{d/2}$ for all sufficiently large $\\lambda$. When $N=2n$ is even and $K$ is a maximal compact subgroup at all places, we prove Weyl's Law for "},"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":"1406.0385","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-02T14:38:42Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"e7ac10e68de2f0b19af1901d213c31179c5eefd369f56303c64cfb192ec64ca3","abstract_canon_sha256":"7f3edb529313c2fa2f907b600e98e07708dd5be7f26cba3091b68477b23a5d3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:41.040832Z","signature_b64":"e+ms2tf6uZQ7sG165ig/GW1xCH1voYZGkSlOJrGTbTrnLrZY96Evuij+qzq4zUnlh18l71yVPmz8IiGwXO8XAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2954ca1144d41fe2844cd2fcf47970be7676948d07eefe04d3331ac075dbe534","last_reissued_at":"2026-05-18T02:50:41.040132Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:41.040132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Density of Self-Dual Automorphic Representations of GL_n(A_Q)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.NT","authors_text":"Vitezslav Kala","submitted_at":"2014-06-02T14:38:42Z","abstract_excerpt":"We study the number $N_{\\mathrm{sd}}^K(\\lambda)$ of self-dual cuspidal automorphic representations of $GL_N(\\mathbb{A_Q})$ which are $K$-spherical with respect to a fixed compact subgroup $K$ and whose Laplacian eigenvalue is $\\leq \\lambda$. We prove Weak Weyl's Law for $N_{\\mathrm{sd}}^K(\\lambda)$ in the form that there are positive constants $c_1, c_2$ (depending on $K$) and $d$ such that $c_1\\lambda^{d/2}\\leq N_{\\mathrm{sd}}^K(\\lambda)\\leq c_2\\lambda^{d/2}$ for all sufficiently large $\\lambda$. When $N=2n$ is even and $K$ is a maximal compact subgroup at all places, we prove Weyl's Law for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0385","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":""},"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":"1406.0385","created_at":"2026-05-18T02:50:41.040243+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.0385v1","created_at":"2026-05-18T02:50:41.040243+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.0385","created_at":"2026-05-18T02:50:41.040243+00:00"},{"alias_kind":"pith_short_12","alias_value":"FFKMUEKE2QP6","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_16","alias_value":"FFKMUEKE2QP6FBCM","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_8","alias_value":"FFKMUEKE","created_at":"2026-05-18T12:28:28.263976+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/FFKMUEKE2QP6FBCM2L6PI6LQXZ","json":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ.json","graph_json":"https://pith.science/api/pith-number/FFKMUEKE2QP6FBCM2L6PI6LQXZ/graph.json","events_json":"https://pith.science/api/pith-number/FFKMUEKE2QP6FBCM2L6PI6LQXZ/events.json","paper":"https://pith.science/paper/FFKMUEKE"},"agent_actions":{"view_html":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ","download_json":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ.json","view_paper":"https://pith.science/paper/FFKMUEKE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.0385&json=true","fetch_graph":"https://pith.science/api/pith-number/FFKMUEKE2QP6FBCM2L6PI6LQXZ/graph.json","fetch_events":"https://pith.science/api/pith-number/FFKMUEKE2QP6FBCM2L6PI6LQXZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ/action/storage_attestation","attest_author":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ/action/author_attestation","sign_citation":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ/action/citation_signature","submit_replication":"https://pith.science/pith/FFKMUEKE2QP6FBCM2L6PI6LQXZ/action/replication_record"}},"created_at":"2026-05-18T02:50:41.040243+00:00","updated_at":"2026-05-18T02:50:41.040243+00:00"}