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Rankin and Swinnerton-Dyer, we prove that for sufficiently large $k,l$, all zeros in the standard fundamental domain are located on the lower boundary $\\mathcal{A} = \\{ e^{i\\theta} : \\pi/2 \\leq \\theta \\leq 2\\pi/3\\}$."},"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":"1907.04259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-09T15:51:57Z","cross_cats_sorted":[],"title_canon_sha256":"a588725dc1a54d485943aedd89054370a6889c7d0bdf26451fc3befd96f0582d","abstract_canon_sha256":"c529be79bf64179168a4c94cdd6ef84890f7e12385d68197ef03dc481f7fb0b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:02.846890Z","signature_b64":"4+D+oHphuGVaBJf15eOwXUnTrOYWe2OXKoa2oJl1Pldo0VJqc2Cfl4tn9o+A43O8kVlA8mkH+3mTtVy3IhSpDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"acbb653d15ad9ad73e9a047da501271d40654706ce9adb9275a580dfdea90f71","last_reissued_at":"2026-05-17T23:41:02.846069Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:02.846069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jetjaroen Klangwang","submitted_at":"2019-07-09T15:51:57Z","abstract_excerpt":"We locate the zeros of the modular forms $E_k^2(\\tau) + E_{2k}(\\tau), E_k^3(\\tau) + E_{3k} (\\tau),$ and $E_k(\\tau)E_l(\\tau) +E_{k+l}(\\tau),$ where $E_k(\\tau)$ is the Eisenstein series for the full modular group $\\text{SL}_2(\\mathbb{Z})$. By utilizing work of F.K.C. Rankin and Swinnerton-Dyer, we prove that for sufficiently large $k,l$, all zeros in the standard fundamental domain are located on the lower boundary $\\mathcal{A} = \\{ e^{i\\theta} : \\pi/2 \\leq \\theta \\leq 2\\pi/3\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04259","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":"1907.04259","created_at":"2026-05-17T23:41:02.846212+00:00"},{"alias_kind":"arxiv_version","alias_value":"1907.04259v1","created_at":"2026-05-17T23:41:02.846212+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04259","created_at":"2026-05-17T23:41:02.846212+00:00"},{"alias_kind":"pith_short_12","alias_value":"VS5WKPIVVWNN","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_16","alias_value":"VS5WKPIVVWNNOPU2","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_8","alias_value":"VS5WKPIV","created_at":"2026-05-18T12:33:30.264802+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/VS5WKPIVVWNNOPU2AR62KAJHDV","json":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV.json","graph_json":"https://pith.science/api/pith-number/VS5WKPIVVWNNOPU2AR62KAJHDV/graph.json","events_json":"https://pith.science/api/pith-number/VS5WKPIVVWNNOPU2AR62KAJHDV/events.json","paper":"https://pith.science/paper/VS5WKPIV"},"agent_actions":{"view_html":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV","download_json":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV.json","view_paper":"https://pith.science/paper/VS5WKPIV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1907.04259&json=true","fetch_graph":"https://pith.science/api/pith-number/VS5WKPIVVWNNOPU2AR62KAJHDV/graph.json","fetch_events":"https://pith.science/api/pith-number/VS5WKPIVVWNNOPU2AR62KAJHDV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV/action/storage_attestation","attest_author":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV/action/author_attestation","sign_citation":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV/action/citation_signature","submit_replication":"https://pith.science/pith/VS5WKPIVVWNNOPU2AR62KAJHDV/action/replication_record"}},"created_at":"2026-05-17T23:41:02.846212+00:00","updated_at":"2026-05-17T23:41:02.846212+00:00"}