{"paper":{"title":"Riccati--Gamma Dynamics for Concavity and Asymptotics of Generalized Dirichlet Eta Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Dragos-Patru Covei","submitted_at":"2026-05-17T15:12:09Z","abstract_excerpt":"We develop a unified analytical and dynamical framework for the qualitative study of the one-parameter family of generalized Dirichlet eta functions $\\eta_{a}(t)=\\sum_{m\\ge 0}(-1)^{m}(am+1)^{-t}$, $a>0$, $t>0$, which specialises to the classical Dirichlet eta and beta functions for $a=1$ and $a=2$. Building on a Mellin--Laplace representation of $\\eta_{a}$ as the expectation $\\mathbb{E}[f_{a}(X_{t})]$ of a scaled logistic function evaluated along a standard Gamma process $(X_{t})_{t\\ge 0}$, we prove that the logarithmic derivative $\\varphi_{a}(t)=\\eta_{a}^{\\prime }(t)/\\eta_{a}(t)$ satisfies a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20238","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/2605.20238/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"}