{"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; the proof assumes standard analytic continuation and zero-free regions or Polya-Vinogradov-type inequalities that are extended to the incomplete mixed setting.","pith_extraction_headline":"For non-principal characters modulo a prime, incomplete mixed sums have maximum size between √p log log p and √p log p."},"references":{"count":15,"sample":[{"doi":"","year":2022,"title":"Moments of polynomials with random multiplicative coefficients","work_id":"f3d48132-e963-4e00-87eb-1310e1b28925","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Distribution of mixed character sums and extremal problems for Littlewood polynomials","work_id":"5233f4e6-fa12-4b8e-ada7-8ce078ac70b0","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Explicit merit factor formulae for Fekete and Turyn polynomials","work_id":"0380785b-5529-48cd-8502-5892ef441669","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"Zeros of Fekete polynomials","work_id":"f6605af0-1b07-45cd-909e-01285ed5ea2f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Upper bounds for theLq norm of Fekete polynomials on subarcs","work_id":"355be989-393b-470f-b981-19b256c20c35","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":15,"snapshot_sha256":"f8c1cbb20fa8b551fbef5ae2e3c0666d39cd31a4cb0e849a3ccbef23a2e697af","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"47d71dbeeac7eebe8a988bc3d086439b650637f8c49cd3f2cfc80ac08dbff360"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}