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Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s, \\pi\\times\\tilde{\\pi})$ and $L(s, \\pi^\\prime\\times\\tilde{\\pi}^\\prime)$, we prove a log-free zero-density estimate for $L(s, \\pi\\times\\pi^\\prime)$ which generalises a result due to Fogels in the context of Dirichlet $L$-functions. We then employ"},"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":"1312.5820","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-12-20T06:25:45Z","cross_cats_sorted":[],"title_canon_sha256":"9419a9e6475073bc864830b53be61bbe4035b12f63bf0a5443bfbe2cfe4254d0","abstract_canon_sha256":"a58414768ed15657d51d212f6dc06d49e896f878f4b048201fe14250a09b5a00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:46.620924Z","signature_b64":"LUXf6hjPormfpzIHsIT1U57e9sMshOmbqkYwur1lIeF43EFscHrqkFzGu9FmH7Y/2qur+X1k6Z4JT7UuapKnCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc1ea4b83d7a106acb90b5c2335f80057f2404046769064cdce615f1a77bd2c8","last_reissued_at":"2026-05-18T02:54:46.620560Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:46.620560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A log-free zero-density estimate and small gaps in coefficients of $L$-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Amir Akbary, Timothy S. Trudgian","submitted_at":"2013-12-20T06:25:45Z","abstract_excerpt":"Let $L(s, \\pi\\times\\pi^\\prime)$ be the Rankin--Selberg $L$-function attached to automorphic representations $\\pi$ and $\\pi^\\prime$. Let $\\tilde{\\pi}$ and $\\tilde{\\pi}^\\prime$ denote the contragredient representations associated to $\\pi$ and $\\pi^\\prime$. Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s, \\pi\\times\\tilde{\\pi})$ and $L(s, \\pi^\\prime\\times\\tilde{\\pi}^\\prime)$, we prove a log-free zero-density estimate for $L(s, \\pi\\times\\pi^\\prime)$ which generalises a result due to Fogels in the context of Dirichlet $L$-functions. 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