{"paper":{"title":"On the mean value of the generalized Dirichlet L-functions with the weight of the Gauss Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Rong Ma, Yana Niu","submitted_at":"2019-12-31T03:13:09Z","abstract_excerpt":"Let $q\\ge3$ be an integer, $\\chi$ denote a Dirichlet character modulo $q$, for any real number $a\\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\\chi,a)=\\sum_{n=1}^{\\infty}\\frac{\\chi(n)}{(n+a)^s}, $$ where $s=\\sigma+it$ with $\\sigma>1$ and $t$ both real. It can be extended to all $s$ by analytic continuation. For any integer $m$, the famous Gauss sum $G(m,\\chi)$ is defined as follows: $$G(m,\\chi)=\\sum_{a=1}^{q}\\chi(a)e\\left(\\frac{am}{q}\\right), $$ where $e(y)=e^{2\\pi iy}$. The main purpose of this paper is to use the analytic method to study the mean value properties of the ge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1912.13153","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/1912.13153/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"}