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In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\\frac{1}{r-1} \\sum_{\\chi \\; \\text{mod} \\; r} |\\sum_{n \\leq x} \\chi(n)| = o(\\sqrt{x})$, and so the sums $\\sum_{n \\leq x} \\chi(n)$ typically exhibit \"better than squareroot cancellation\". We prove analogous better than squareroot bounds for the moments $\\frac{1}{T} \\int_{0}^{T} |\\sum_{n \\leq x} n^"},"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":"2301.04390","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2023-01-11T10:28:14Z","cross_cats_sorted":[],"title_canon_sha256":"08cd3fc4eb1285df4320b4e5688ce410e6599188497e14946590fbdabb81ef4b","abstract_canon_sha256":"50d64c569e27681217d3fb6136e344d8849808f6ce5320801c78bb3cb3c8dfb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:32:25.049379Z","signature_b64":"sq9W10MMh2zvsclom4R5dTjcAAqmqQKMaunaB8fc143fmYNfAS/PV88LhrxuFLDrjedO3Thg8yIDph9TowxEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"937e4cf5ba01c1f188317446fa77d88ed54eda54645b61d8d9e822c8b28ae08f","last_reissued_at":"2026-07-05T05:32:25.048956Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:32:25.048956Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The typical size of character and zeta sums is $o(\\sqrt{x})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam J. Harper","submitted_at":"2023-01-11T10:28:14Z","abstract_excerpt":"We prove conjecturally sharp upper bounds for the Dirichlet character moments $\\frac{1}{r-1} \\sum_{\\chi \\; \\text{mod} \\; r} |\\sum_{n \\leq x} \\chi(n)|^{2q}$, where $r$ is a large prime, $1 \\leq x \\leq r$, and $0 \\leq q \\leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\\frac{1}{r-1} \\sum_{\\chi \\; \\text{mod} \\; r} |\\sum_{n \\leq x} \\chi(n)| = o(\\sqrt{x})$, and so the sums $\\sum_{n \\leq x} \\chi(n)$ typically exhibit \"better than squareroot cancellation\". 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