Abstract
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) = Ʃn≤xa(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.
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Acknowledgements
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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Li, J.J., Wang, P.W. & Zhang, M. On the fourth power moment of Fourier coefficients of cusp form. Acta. Math. Sin.-English Ser. 34, 1050–1058 (2018). https://doi.org/10.1007/s10114-017-6508-5
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DOI: https://doi.org/10.1007/s10114-017-6508-5