Acta Mathematica Sinica, English Series

, Volume 34, Issue 6, pp 1050–1058 | Cite as

On the fourth power moment of Fourier coefficients of cusp form



Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A(x) = Ʃnxa(n). In this paper, we establish an asymptotic formula of the fourth power moment of A(x) and prove that \(\int_1^T {{A^4}\left( x \right)dx = \frac{3}{{64\kappa {\pi ^4}}}{s_{4;2}}\left( {\tilde a} \right){T^{2\kappa }} + O\left( {{T^{2\kappa - {\delta _4} + \varepsilon }}} \right)} \) with δ4 = 1/8, which improves the previous result.


Cusp form Fourier coefficient mean value asymptotic formula 

MR(2010) Subject Classification

11N37 11M06 11P21 


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The authors would like to express the most and the greatest sincere gratitude to Professor Wenguang Zhai for his valuable advices and constant encouragement.


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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsChina University of Mining and TechnologyBeijingP. R. China

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