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We show that if $\\tau_0$ is an imaginary quadratic number with $\\mathrm{Im}(\\tau_0)>0$ and $m$ is an odd integer, then $\\sqrt{m}\\varphi(m\\tau_0)/\\varphi(\\tau_0)$ is an algebraic integer dividing $\\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and"},"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":"1008.0473","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-03T07:29:52Z","cross_cats_sorted":[],"title_canon_sha256":"630c74687b18c8523f3bddbe7c8afea86fbd2146ffec5cf8824101d7ab384244","abstract_canon_sha256":"b6ea7919a77708cbd1a1896b8289251b57c8368d788bda68182e686c3fb0315b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:33.470623Z","signature_b64":"ST4De9SgWaKf5lZu/2rt7wxZEXV83l7PP80ofxbdm4fkoLDvNIhThER1q/B4W4R/H3MfHBfM7pKYVoKHZLijDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3975aec5e0f89769c0c4f3dba914f53687eef239192aed3b236c4dbedb8e7570","last_reissued_at":"2026-05-18T04:42:33.470012Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:33.470012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic integers as special values of modular units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dong Hwa Shin, Dong Sung Yoon, Ja Kyung Koo","submitted_at":"2010-08-03T07:29:52Z","abstract_excerpt":"Let $\\varphi(\\tau)=\\eta((\\tau+1)/2)^2/\\sqrt{2\\pi}e^\\frac{\\pi i}{4}\\eta(\\tau+1)$ where $\\eta(\\tau)$ is the Dedekind eta-function. We show that if $\\tau_0$ is an imaginary quadratic number with $\\mathrm{Im}(\\tau_0)>0$ and $m$ is an odd integer, then $\\sqrt{m}\\varphi(m\\tau_0)/\\varphi(\\tau_0)$ is an algebraic integer dividing $\\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0473","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":"1008.0473","created_at":"2026-05-18T04:42:33.470099+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.0473v2","created_at":"2026-05-18T04:42:33.470099+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.0473","created_at":"2026-05-18T04:42:33.470099+00:00"},{"alias_kind":"pith_short_12","alias_value":"HF225RPA7CLW","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_16","alias_value":"HF225RPA7CLWTQGE","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_8","alias_value":"HF225RPA","created_at":"2026-05-18T12:26:07.630475+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/HF225RPA7CLWTQGE6PN2SFHVG2","json":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2.json","graph_json":"https://pith.science/api/pith-number/HF225RPA7CLWTQGE6PN2SFHVG2/graph.json","events_json":"https://pith.science/api/pith-number/HF225RPA7CLWTQGE6PN2SFHVG2/events.json","paper":"https://pith.science/paper/HF225RPA"},"agent_actions":{"view_html":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2","download_json":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2.json","view_paper":"https://pith.science/paper/HF225RPA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.0473&json=true","fetch_graph":"https://pith.science/api/pith-number/HF225RPA7CLWTQGE6PN2SFHVG2/graph.json","fetch_events":"https://pith.science/api/pith-number/HF225RPA7CLWTQGE6PN2SFHVG2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2/action/storage_attestation","attest_author":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2/action/author_attestation","sign_citation":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2/action/citation_signature","submit_replication":"https://pith.science/pith/HF225RPA7CLWTQGE6PN2SFHVG2/action/replication_record"}},"created_at":"2026-05-18T04:42:33.470099+00:00","updated_at":"2026-05-18T04:42:33.470099+00:00"}