{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:GY423AXOJXB2FZNFXHU2DPLQVO","short_pith_number":"pith:GY423AXO","schema_version":"1.0","canonical_sha256":"3639ad82ee4dc3a2e5a5b9e9a1bd70ab96f36fdb9266f4ef8eeee3d2a4475959","source":{"kind":"arxiv","id":"2409.13436","version":1},"attestation_state":"computed","paper":{"title":"A lower bound on high moments of character sums","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Barnab\\'as Szab\\'o","submitted_at":"2024-09-20T11:59:43Z","abstract_excerpt":"For any real $k\\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum\n  \\begin{equation*}\n  \\frac{1}{\\phi(q)} \\sum_{\\substack{\\chi \\text{ mod }q\\\\ \\chi\\neq \\chi_0}} \\Big| \\sum_{n\\leq x} \\chi(n)\\Big|^{2k}, \\end{equation*} where $1\\leq x\\leq q$, and $\\chi$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method."},"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":"2409.13436","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-09-20T11:59:43Z","cross_cats_sorted":[],"title_canon_sha256":"e2abb9bf9196be8a384f17a29bba87c9e90893fc83f16fb9809b3c14b7801963","abstract_canon_sha256":"cbda14decdbd39126258806acd4adf12cb523fdb6cd72b391d7f45ac60b2024c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:09:35.372670Z","signature_b64":"R314ux44gna562gkGmp4edw9d4lRWg05YlpPKhAf7lQuvQUvSVS3tm5tkNmci41JbZCqeth2dmNLmwWVsZuUCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3639ad82ee4dc3a2e5a5b9e9a1bd70ab96f36fdb9266f4ef8eeee3d2a4475959","last_reissued_at":"2026-07-05T09:09:35.372091Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:09:35.372091Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A lower bound on high moments of character sums","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Barnab\\'as Szab\\'o","submitted_at":"2024-09-20T11:59:43Z","abstract_excerpt":"For any real $k\\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum\n  \\begin{equation*}\n  \\frac{1}{\\phi(q)} \\sum_{\\substack{\\chi \\text{ mod }q\\\\ \\chi\\neq \\chi_0}} \\Big| \\sum_{n\\leq x} \\chi(n)\\Big|^{2k}, \\end{equation*} where $1\\leq x\\leq q$, and $\\chi$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.13436","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.13436/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2409.13436","created_at":"2026-07-05T09:09:35.372158+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.13436v1","created_at":"2026-07-05T09:09:35.372158+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.13436","created_at":"2026-07-05T09:09:35.372158+00:00"},{"alias_kind":"pith_short_12","alias_value":"GY423AXOJXB2","created_at":"2026-07-05T09:09:35.372158+00:00"},{"alias_kind":"pith_short_16","alias_value":"GY423AXOJXB2FZNF","created_at":"2026-07-05T09:09:35.372158+00:00"},{"alias_kind":"pith_short_8","alias_value":"GY423AXO","created_at":"2026-07-05T09:09:35.372158+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2607.01184","citing_title":"Lower bounds for low moments of character sums, I: Short sums with general multiplicative weights","ref_index":19,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO","json":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO.json","graph_json":"https://pith.science/api/pith-number/GY423AXOJXB2FZNFXHU2DPLQVO/graph.json","events_json":"https://pith.science/api/pith-number/GY423AXOJXB2FZNFXHU2DPLQVO/events.json","paper":"https://pith.science/paper/GY423AXO"},"agent_actions":{"view_html":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO","download_json":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO.json","view_paper":"https://pith.science/paper/GY423AXO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.13436&json=true","fetch_graph":"https://pith.science/api/pith-number/GY423AXOJXB2FZNFXHU2DPLQVO/graph.json","fetch_events":"https://pith.science/api/pith-number/GY423AXOJXB2FZNFXHU2DPLQVO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO/action/storage_attestation","attest_author":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO/action/author_attestation","sign_citation":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO/action/citation_signature","submit_replication":"https://pith.science/pith/GY423AXOJXB2FZNFXHU2DPLQVO/action/replication_record"}},"created_at":"2026-07-05T09:09:35.372158+00:00","updated_at":"2026-07-05T09:09:35.372158+00:00"}