{"paper":{"title":"Hardy-Littlewood-Riesz type equivalent criteria for the Generalized Riemann hypothesis","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bibekananda Maji, Meghali Garg","submitted_at":"2022-08-16T08:19:36Z","abstract_excerpt":"In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet $L$-function $L(s,\\chi)$ is equivalent to the following bound: Let $k \\geq 1$ and $\\ell$ be positive real numbers. For any $\\epsilon >0$, we have \\begin{align*} \\sum_{n=1}^{\\infty} \\frac{\\chi(n) \\mu(n)}{n^{k}} \\exp \\left(- \\frac{ x}{n^{\\ell}}\\right) = O_{\\epsilon,k,\\ell} \\bigg(x^{-\\frac{k}{\\ell}+\\frac{1}{2 \\ell} + \\epsilon }\\bigg), \\quad \\mathrm{as}\\,\\, x \\rightarrow \\infty, \\end{align*} where $\\chi$ is a primitive Dirichlet character modulo $q$, and $\\mu(n)$ denotes the M\\\"{o}bius function. This bound gen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.07596","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/2208.07596/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"}