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The distribution of Fourier coefficients of cusp forms over sparse sequences

Volume 163 / 2014

Huixue Lao, Ayyadurai Sankaranarayanan Acta Arithmetica 163 (2014), 101-110 MSC: Primary 11F30; Secondary 11F66. DOI: 10.4064/aa163-2-1


Let $\lambda_f(n)$ be the $n$th normalized Fourier coefficient of a holomorphic Hecke eigenform $f(z)\in S_{k}(\Gamma)$. We establish that $\sum_{n \leq x}\lambda_f^2(n^j)=c_{j} x+O(x^{1-\frac{2}{(j+1)^2+1}})$ for $j=2,3,4,$ which improves the previous results. For $j=2$, we even establish a better result.


  • Huixue LaoDepartment of Mathematics
    Shandong Normal University
    250014 Jinan, China
  • Ayyadurai SankaranarayananSchool of Mathematics
    Tata Institute of Fundamental Research
    400005 Mumbai, India

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