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Integral mean square estimation for the error term related to \(\sum\nolimits_{n \leqslant x} {\lambda ^2 (n^2 )} \)

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Abstract

Let λ f (n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ S k (Γ). We establish that, for any ɛ > 0,

$\frac{1} {X}\int_1^X {\left| {\sum\limits_{n \leqslant x} {\lambda _f^2 (n^2 ) - c_2 x} } \right|^2 dx \ll _\varepsilon X\tfrac{{154}} {{101}} + \varepsilon } , $

which improves previous results.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China

    HuiXue Lao

  2. School of Mathematics, Tata Institute of Fundamental Research, Mumbai, 400005, India

    Ayyadurai Sankaranarayanan

Authors
  1. HuiXue Lao
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  2. Ayyadurai Sankaranarayanan
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Correspondence to Ayyadurai Sankaranarayanan.

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Lao, H., Sankaranarayanan, A. Integral mean square estimation for the error term related to \(\sum\nolimits_{n \leqslant x} {\lambda ^2 (n^2 )} \) . Sci. China Math. 58, 1–8 (2015). https://doi.org/10.1007/s11425-015-5011-7

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