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Motivated by the problem of the first sign change of $\\lambda_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\\sum_{n\\leq x} \\lambda_f(n)$. We first show that $S_f(x)=o(x\\log x)$ implies that $\\lambda_f(n)<0$ for some $n\\leq x$. We also prove that $S_f(x)=o(x\\log x)$ in the range $\\log x/\\log\\log k\\to \\infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore "},"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":"1703.10582","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-30T17:30:23Z","cross_cats_sorted":[],"title_canon_sha256":"80d084a936cb983b02759c9571c2cebdae365bc90b1309cf71be4cdebf1cb6b4","abstract_canon_sha256":"61649778556b4f1a740f96cd59769760b3665b93701c968d137097d07776128b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:34.353287Z","signature_b64":"EszbvBAyo90a3HP0B9gpKQI9yb3eKdOHQ1wPNFdLUSDbgYu5B64RtCmqtrkKwsp60+tn/2+r5Erl+R618pzeDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02ffec35947ce26688bcafcf4b86c64d91200f84ff274f969f2600d992c38e19","last_reissued_at":"2026-05-18T00:47:34.352772Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:34.352772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large sums of Hecke eigenvalues of holomorphic cusp forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Youness Lamzouri","submitted_at":"2017-03-30T17:30:23Z","abstract_excerpt":"Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\\{\\lambda_f(n)\\}_{n\\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\\lambda_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\\sum_{n\\leq x} \\lambda_f(n)$. We first show that $S_f(x)=o(x\\log x)$ implies that $\\lambda_f(n)<0$ for some $n\\leq x$. We also prove that $S_f(x)=o(x\\log x)$ in the range $\\log x/\\log\\log k\\to \\infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10582","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":"1703.10582","created_at":"2026-05-18T00:47:34.352836+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.10582v1","created_at":"2026-05-18T00:47:34.352836+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.10582","created_at":"2026-05-18T00:47:34.352836+00:00"},{"alias_kind":"pith_short_12","alias_value":"AL76YNMUPTRG","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_16","alias_value":"AL76YNMUPTRGNCF4","created_at":"2026-05-18T12:31:05.417338+00:00"},{"alias_kind":"pith_short_8","alias_value":"AL76YNMU","created_at":"2026-05-18T12:31:05.417338+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/AL76YNMUPTRGNCF4V7HUXBWGJW","json":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW.json","graph_json":"https://pith.science/api/pith-number/AL76YNMUPTRGNCF4V7HUXBWGJW/graph.json","events_json":"https://pith.science/api/pith-number/AL76YNMUPTRGNCF4V7HUXBWGJW/events.json","paper":"https://pith.science/paper/AL76YNMU"},"agent_actions":{"view_html":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW","download_json":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW.json","view_paper":"https://pith.science/paper/AL76YNMU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.10582&json=true","fetch_graph":"https://pith.science/api/pith-number/AL76YNMUPTRGNCF4V7HUXBWGJW/graph.json","fetch_events":"https://pith.science/api/pith-number/AL76YNMUPTRGNCF4V7HUXBWGJW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW/action/storage_attestation","attest_author":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW/action/author_attestation","sign_citation":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW/action/citation_signature","submit_replication":"https://pith.science/pith/AL76YNMUPTRGNCF4V7HUXBWGJW/action/replication_record"}},"created_at":"2026-05-18T00:47:34.352836+00:00","updated_at":"2026-05-18T00:47:34.352836+00:00"}