{"paper":{"title":"Negative discrete second moments of Dirichlet $L$-functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Pearce-Crump","submitted_at":"2026-06-23T18:59:58Z","abstract_excerpt":"Let $\\chi$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,\\chi)$ and that the non-trivial zeros $\\rho=\\tfrac12+i\\gamma$ of $L(s,\\chi)$ are simple, we prove lower bounds for the discrete moments $\\sum_{0<\\gamma\\le T}|L'(\\rho,\\chi)|^{-2}$ and $\\sum_{0<\\gamma\\le T}|L(2\\rho,\\chi^2)/L'(\\rho,\\chi)|^2$, uniformly in the conductor. The bounds capture the proportion $\\beta/(1+\\beta)$ of the conjectured asymptotics, where $\\beta=\\log T/\\log qT$: this is one half whenever $\\log q=o(\\log T)$, recovering for fixed $q$ the Dirichlet analogues of theorem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25094","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/2606.25094/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"}