{ "id": "2502.01288", "version": "v1", "published": "2025-02-03T11:59:20.000Z", "updated": "2025-02-03T11:59:20.000Z", "title": "On an unconditional $\\rm GL_3$ analog of Selberg's result", "authors": [ "Qingfeng Sun", "Hui Wang" ], "comment": "21 pages. Comments welcome!", "categories": [ "math.NT" ], "abstract": "Let $S_F(t)=\\pi^{-1}\\arg L(1/2+it, F)$, where $F$ is a Hecke--Maass cusp form for $\\rm SL_3(\\mathbb{Z})$ in the generic position with the spectral parameter $\\nu_{F}=\\big(\\nu_{F,1},\\nu_{F,2},\\nu_{F,3}\\big)$ and the Langlands parameter $\\mu_{F}=\\big(\\mu_{F,1},\\mu_{F,2},\\mu_{F,3}\\big)$. In this paper, we establish an unconditional asymptotic formula for the moments of $S_F(t)$. Previouly, such a formula was only known under the Generalized Riemann Hypothesis. The key ingredient is a weighted zero-density estimate in the spectral aspect for $L(s, F)$ which was recently proved by the authors in [18].", "revisions": [ { "version": "v1", "updated": "2025-02-03T11:59:20.000Z" } ], "analyses": { "subjects": [ "11F12", "11F66", "11F72" ], "keywords": [ "selbergs result", "hecke-maass cusp form", "unconditional asymptotic formula", "weighted zero-density estimate", "generic position" ], "note": { "typesetting": "TeX", "pages": 21, "language": "en", "license": "arXiv", "status": "editable" } } }