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<p>Let <i>F</i> be a Hecke-Maass cusp form for SL<sub>3</sub>(ℤ) with the Langlands parameter <i>μ</i><sub><i>F</i></sub> = (<i>μ</i><sub><i>F</i>,1</sub>, <i>μ</i><sub><i>F</i>,2</sub>, <i>μ</i><sub><i>F</i>,3</sub>) and the associated <i>L</i>-function <i>L</i>(<i>s</i>, <i>F</i>). Define <i>S</i><sub><i>F</i></sub>(<i>t</i>) = <i>π</i><sup>−1</sup> arg <i>L</i>(1/2 + i<i>t</i>, <i>F</i>). When <i>μ</i><sub><i>F</i></sub> is in generic position, we establish an unconditional asymptotic formula for the moments of <i>S</i><sub><i>F</i></sub>(<i>t</i>). Previously, such a formula was only known to hold under the generalized Riemann hypothesis. The key ingredient is a weighted zero-density estimate in the spectral aspect for <i>L</i>(<i>s</i>, <i>F</i>), which has recently been proved by Sun and Wang (arXiv:2412.02416, 2024).</p>

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On an unconditional GL3 analog of Selberg’s result

  • 期刊论文

  • 发表日期：

    2026年4月21日

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  • Qingfeng Sun,
  • Hui Wang

摘要

Let F be a Hecke-Maass cusp form for SL3(ℤ) with the Langlands parameter μF = (μF,1, μF,2, μF,3) and the associated L-function L(s, F). Define SF(t) = π−1 arg L(1/2 + it, F). When μF is in generic position, we establish an unconditional asymptotic formula for the moments of SF(t). Previously, such a formula was only known to hold under the generalized Riemann hypothesis. The key ingredient is a weighted zero-density estimate in the spectral aspect for L(s, F), which has recently been proved by Sun and Wang (arXiv:2412.02416, 2024).