{"paper":{"title":"Note on the $a$-points of the Riemann zeta function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min-Jie Luo, Peng-Cheng Hang","submitted_at":"2024-11-20T12:14:02Z","abstract_excerpt":"For any $a\\in\\mathbb{C}$, the zeros of $\\zeta(s)-a$, denoted by $\\rho_a=\\beta_a+i\\gamma_a$, are called $a$-points of the Riemann zeta function $\\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\\zeta(s)$ shown by Garunk\\v{s}tis and Steuding. We then deduce an asymptotic of the sum \\[S_T(a,\\delta)=\\sum_{\\tau<\\gamma_a\\leqslant T}\\zeta'(\\rho_a+i\\delta)X^{\\rho_a},\\quad T\\to\\infty,\\] where $0\\ne\\delta=\\frac{2\\pi\\alpha}{\\log\\frac{T}{2\\pi X}}\\ll 1$, and $X>0$ and $\\tau\\geqslant|\\delta|+1$ are fixed.\n  We also find the interesting varied behavior of $S_T(a,\\delta)$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13255","kind":"arxiv","version":2},"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/2411.13255/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"}