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We prove that $$ \\sqrt p \\log \\log p \\ll \\max_{0 \\le \\theta < 1}{\\left|F_\\chi(\\alpha,\\beta;\\theta)\\right|} \\ll \\sqrt{p}\\log p, $$ generalizing an old result of Montgomery as well as a recent result of Iggidr in two aspects: we allow general non-principal characters $\\chi$, and we consider incomplete mixed character sums."},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.13715","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T16:00:49Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"a77cbed095917448ea430c685516d7a1d4f98649f63cbed9dbafc1d2f65ea4f2","abstract_canon_sha256":"7c5e4c0f3ed51a3c0686beae410510a32c295a8b6ddfa0623aa02c6eefc5a5cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:16.713417Z","signature_b64":"ab3xq0Y4aFRat6oCsM6FNELPYLehudMO4fDPKfit5IHmMvBdNUX/8LHl5086QzMM1BlFqMK+2axQp3wcb+q2DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"305bd8d4e60a057356aae70a1c97bf618664bee634f1f9e990adab9d05f39995","last_reissued_at":"2026-05-18T02:44:16.712886Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:16.712886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large values of shifted mixed character sums","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"For non-principal characters modulo a prime, incomplete mixed sums have maximum size between √p log log p and √p log p.","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"N\\'eo Tardy","submitted_at":"2026-05-13T16:00:49Z","abstract_excerpt":"We consider sums of the form $$F_\\chi(\\alpha,\\beta;\\theta) := \\sum_{\\alpha p<n\\le\\beta p}\\chi(n)e(n\\theta),$$ where $\\chi$ is a non-principal Dirichlet character modulo a prime number $p$. We prove that $$ \\sqrt p \\log \\log p \\ll \\max_{0 \\le \\theta < 1}{\\left|F_\\chi(\\alpha,\\beta;\\theta)\\right|} \\ll \\sqrt{p}\\log p, $$ generalizing an old result of Montgomery as well as a recent result of Iggidr in two aspects: we allow general non-principal characters $\\chi$, and we consider incomplete mixed character sums."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"√p log log p ≪ max_{0 ≤ θ < 1} |F_χ(α,β;θ)| ≪ √p log p for non-principal χ mod prime p.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The character χ is non-principal and p is an odd prime; the proof assumes standard analytic continuation and zero-free regions or Polya-Vinogradov-type inequalities that are extended to the incomplete mixed setting.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For non-principal χ mod prime p, the max over θ of |sum_{αp < n ≤ βp} χ(n) e(nθ)| satisfies √p log log p ≪ max ≪ √p log p.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For non-principal characters modulo a prime, incomplete mixed sums have maximum size between √p log log p and √p log p.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9d92571358cda831e1bb463ad28fa7a2721c0d93d64ca216d4dba47ea6913be4"},"source":{"id":"2605.13715","kind":"arxiv","version":1},"verdict":{"id":"7a8ddc7a-bb9d-494f-8358-7dda5ad91fc6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:43:37.280913Z","strongest_claim":"√p log log p ≪ max_{0 ≤ θ < 1} |F_χ(α,β;θ)| ≪ √p log p for non-principal χ mod prime p.","one_line_summary":"For non-principal χ mod prime p, the max over θ of |sum_{αp < n ≤ βp} χ(n) e(nθ)| satisfies √p log log p ≪ max ≪ √p log p.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The character χ is non-principal and p is an odd prime; 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