Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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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.

$\Omega$-result on coefficients of automorphic $L$-functions over sparse sequences

Huixue Lao and Hongbin Wei

Shandong Normal University, Shandong Normal University

Abstract

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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