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<p>Let ⋋ <sub><i>f</i></sub> (<i>n</i>), <i>σ</i>(<i>n</i>), and <i>ϕ</i> (<i>n</i>) denote the <i>n</i>th Fourier coefficient of a holomorphic cusp form <i>f</i>, the divisor function, and Euler’s totient function, respectively. In this paper, we refine the error term in the asymptotic formula for the sum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\sum }_{n\le x}{\leftthreetimes}_{f}^{j}\left(n\right){\sigma }^{b}\left(n\right){\phi }^{c}\left(n\right),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msubsup> <mo>⋋</mo> <mrow> <mi>f</mi> </mrow> <mi>j</mi> </msubsup> <mfenced close=")" open="("> <mi>n</mi> </mfenced> <msup> <mrow> <mi>σ</mi> </mrow> <mi>b</mi> </msup> <mfenced close=")" open="("> <mi>n</mi> </mfenced> <msup> <mrow> <mi>ϕ</mi> </mrow> <mi>c</mi> </msup> <mfenced close=")" open="("> <mi>n</mi> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <i>j</i> is a positive integer.</p>

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Mean value estimates of hybrid arithmetic functions involving the Fourier coefficients of cusp forms

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • Yunlong Liu,
  • Xiaofei Yan

摘要

Let ⋋ f (n), σ(n), and ϕ (n) denote the nth Fourier coefficient of a holomorphic cusp form f, the divisor function, and Euler’s totient function, respectively. In this paper, we refine the error term in the asymptotic formula for the sum \({\sum }_{n\le x}{\leftthreetimes}_{f}^{j}\left(n\right){\sigma }^{b}\left(n\right){\phi }^{c}\left(n\right),\) ∑ n ≤ x ⋋ f j n σ b n ϕ c n , where j is a positive integer.