J. Korean Math. Soc. 2015; 52(5): 945-954
Printed September 1, 2015
https://doi.org/10.4134/JKMS.2015.52.5.945
Copyright © The Korean Mathematical Society.
Huixue Lao and Hongbin Wei
Shandong Normal University, Shandong Normal University
Let $\lambda_{f}(n)$ denote the $n$-th normalized Fourier coefficient of a primitive holomorphic form $f$ for the full modular group $\Gamma=SL_{2}(\mathbb{Z})$. In this paper, we are concerned with $\Omega$-result on the summatory function $\sum_{n\leqslant x}\lambda^{2}_f(n^{2}),$ and establish the following result $$\sum_{n\leqslant x}\lambda_{f}^{2}(n^{2}) =c_1 x + \Omega(x^{\frac{4}{9}}),$$ where $c_1$ is a suitable constant.
Keywords: automorphic $L$-functions, holomorphic cusp forms, Omega theorem
MSC numbers: 11F30, 11F66
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