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On sums involving coefficients of automorphic $ L$-functions

Author: Guangshi Lü
Journal: Proc. Amer. Math. Soc. 137 (2009), 2879-2887
MSC (2000): Primary 11F30, 11F11, 11F66
Published electronically: March 27, 2009
MathSciNet review: 2506445
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Abstract: Let $ L(s,\pi)$ be the automorphic $ L$-function associated to an automorphic irreducible cuspidal representation $ \pi$ of GL$ _m$ over $ \mathbb{Q}$, and let $ a_{\pi}(n)$ be the $ n$th coefficient in its Dirichlet series expansion. In this paper we prove that if at every finite place $ p$, $ \pi_p$ is unramified, then for any $ \varepsilon>0$,

$\displaystyle A_{\pi}(x)=\sum_{n \leq x}a_{\pi}(n) \ll_{\varepsilon,\pi} \left\... ... x^{\frac{m^2-m}{m^2+1}+\varepsilon}& \text{ if }& m \geq 3. \end{array}\right.$      

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Additional Information

Guangshi Lü
Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China

Keywords: Automorphic $L$-function, symmetric square $L$-function, Ramanujan conjecture
Received by editor(s): December 1, 2008
Published electronically: March 27, 2009
Additional Notes: This work was supported by the National Natural Science Foundation of China (Grant No. 10701048)
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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