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<p>Let <i>S</i>(<i>t, f</i>) = <i>π</i><sup>-1</sup> arg<i>L</i>(1<i>/</i>2 + i<i>t, f</i>), where <i>f</i> is a holomorphic Hecke cusp form of weight 2 and prime level <i>q</i>. In this paper, we establish an unconditional asymptotic formula for the moments of <i>S</i>(<i>t, f</i>), providing a level aspect analogue of Selberg’s classical work on <i>S</i>(<i>t</i>). As a consequence, we derive a weighted central limit theorem for the distribution of <i>S</i>(<i>t, f</i>) normalized by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sqrt{\text{loglog}q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mrow> <mtext>loglog</mtext> <mi>q</mi> </mrow> </msqrt> </math></EquationSource> </InlineEquation>. To this end, we develop a precise approximation for <i>S</i>(<i>t, f</i>) via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of <i>L</i>-functions.</p>

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On a level analog of Selberg’s result on S(t)

  • 期刊论文

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  • 发表日期：

    2026年3月6日

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  • Qingfeng Sun,
  • Hui Wang

摘要

Let S(t, f) = π-1 argL(1/2 + it, f), where f is a holomorphic Hecke cusp form of weight 2 and prime level q. In this paper, we establish an unconditional asymptotic formula for the moments of S(t, f), providing a level aspect analogue of Selberg’s classical work on S(t). As a consequence, we derive a weighted central limit theorem for the distribution of S(t, f) normalized by \(\sqrt{\text{loglog}q}\) loglog q . To this end, we develop a precise approximation for S(t, f) via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of L-functions.