{"paper":{"title":"Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For large prime P, the sum over characters mod P of low-lying zeros of L(s, chi) on the critical line in intervals of length T is at least order T squared times P times sqrt(log P), even for T as small as 1 over sqrt(log P).","cross_cats":[],"primary_cat":"math.NT","authors_text":"XinHang Ji","submitted_at":"2026-05-10T03:14:41Z","abstract_excerpt":"In this paper, we establish a new lower bound for the number of low-lying zeros of Dirichlet $L$-functions $L(s, \\chi)$ on the critical line within extremely short intervals. Specifically, for a sufficiently large prime $P$ and real number $T \\in [a_1/\\sqrt{\\log P}, 1]$, we prove that the sum of the number of zeros on the critical line $N_0(T, \\chi)$ over characters $\\chi \\bmod P$ satisfies $$ \\sum_{\\chi \\bmod P} N_0(T, \\chi) \\gg T^2 P\\sqrt{\\log P} .$$ Traditional approaches encounter significant technical barriers in this short-interval regime. The Levinson method fails due to its own inheren"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for a sufficiently large prime P and real number T in [a1/sqrt(log P), 1], we prove that sum_{chi mod P} N0(T, chi) >> T^2 P sqrt(log P)","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The high-dimensional Mellin transform framework can be applied to the multi-variable series from the mollifier without residual cross-terms or error terms that would invalidate the lower bound extraction in the stated short-interval range.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For large prime P and T at least on the order of 1 over sqrt(log P), the summed count of low-lying zeros on the critical line over characters mod P satisfies sum N0(T, chi) much greater than T squared P sqrt(log P).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For large prime P, the sum over characters mod P of low-lying zeros of L(s, chi) on the critical line in intervals of length T is at least order T squared times P times sqrt(log P), even for T as small as 1 over sqrt(log P).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"19dff9ef54e8ed64aa2d95f1ddc47303b83df52a3e55a86c7cac9f8af176eadd"},"source":{"id":"2605.09282","kind":"arxiv","version":2},"verdict":{"id":"be485864-16fd-4594-ae95-3c41f0828950","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T03:54:31.609772Z","strongest_claim":"for a sufficiently large prime P and real number T in [a1/sqrt(log P), 1], we prove that sum_{chi mod P} N0(T, chi) >> T^2 P sqrt(log P)","one_line_summary":"For large prime P and T at least on the order of 1 over sqrt(log P), the summed count of low-lying zeros on the critical line over characters mod P satisfies sum N0(T, chi) much greater than T squared P sqrt(log P).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The high-dimensional Mellin transform framework can be applied to the multi-variable series from the mollifier without residual cross-terms or error terms that would invalidate the lower bound extraction in the stated short-interval range.","pith_extraction_headline":"For large prime P, the sum over characters mod P of low-lying zeros of L(s, chi) on the critical line in intervals of length T is at least order T squared times P times sqrt(log P), even for T as small as 1 over sqrt(log P)."},"integrity":{"clean":false,"summary":{"advisory":1,"critical":1,"by_detector":{"doi_compliance":{"total":2,"advisory":1,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.09282/integrity.json","findings":[{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.1090/S0273-0979-99-00752-928) was visible in the surrounding text but could not be confirmed against doi.org as printed.","detector":"doi_compliance","severity":"advisory","ref_index":5,"audited_at":"2026-05-19T10:22:14.106597Z","detected_doi":"10.1090/S0273-0979-99-00752-928","finding_type":"recoverable_identifier","verdict_class":"incontrovertible","detected_arxiv_id":null},{"note":"Identifier '10.1007/s40993-022-00361-9' is syntactically valid but the DOI registry (doi.org) returned 404, and Crossref / OpenAlex / internal corpus also have no record. The cited work could not be located through any authoritative source.","detector":"doi_compliance","severity":"critical","ref_index":8,"audited_at":"2026-05-19T10:22:14.106597Z","detected_doi":"10.1007/s40993-022-00361-9","finding_type":"unresolvable_identifier","verdict_class":"cross_source","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T20:34:13.864334Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T13:31:17.746202Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:22:14.106597Z","status":"completed","version":"1.0.0","findings_count":2}],"snapshot_sha256":"b37216b92d991c08b48f1b96b73efe457bda8ff8abfe743ba0b5dc4f324cbf07"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"c9f96398574b0a366f2a6fa5dc66b1626881085fb0cde55ac1b0da4e7a06b01f"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}