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Sums of Fourier coefficients of cusp forms of level $ D$ twisted by exponential functions over arithmetic progressions


Authors: Huan Liu and Meng Zhang
Journal: Proc. Amer. Math. Soc. 145 (2017), 3761-3774
MSC (2010): Primary 11L07, 11F30; Secondary 11B25
DOI: https://doi.org/10.1090/proc/13536
Published electronically: April 27, 2017
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Abstract: Let $ g$ be a holomorphic Hecke newform of level $ D$ and $ \lambda _{g}(n)$ be its $ n$-th Fourier coefficient. We prove that the sum $ \mathcal {S}_{D}(N,\alpha , \beta ,X)=\sum _{\substack {X<n\leq 2X\\ n\equiv l \,\text {\textrm {mod}} N}}\lambda _{g}(n)e(\alpha n^\beta )$ has an asymptotic formula for the case of $ \beta =1/2$, $ \alpha $ close to $ \pm 2\sqrt {q/c^2D_2}$, where $ l$, $ q$, $ c$, $ D_2$ are positive integers satisfying $ (l,N)=1$, $ c\vert N$, $ D_2=D/(c, D)$ and $ X$ is sufficiently large. We obtain upper bounds of $ \mathcal {S}_{D}(N,\alpha , \beta ,X)$ for the case of $ 0<\beta <1$ and $ \alpha \in \mathbb{R}$.


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Additional Information

Huan Liu
Affiliation: School of Mathematics, Shandong University, 27 Shanda Nanlu Jinan, Shandong 250100, People’s Republic of China
Email: liuhuansdu@hotmail.com

Meng Zhang
Affiliation: School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, 40 Shungeng Road Jinan, Shandong 250014, People’s Republic of China
Email: zhmengsdu@hotmail.com

DOI: https://doi.org/10.1090/proc/13536
Keywords: Exponential sums, cusp form for $GL_2$, arithmetic progressions
Received by editor(s): June 2, 2016
Received by editor(s) in revised form: July 30, 2016, August 26, 2016, and October 14, 2016
Published electronically: April 27, 2017
Additional Notes: The authors would like to express their thanks to the referee for the careful reading of the paper and valuable suggestions.
The authors were supported by the National Natural Science Foundation of China (Grant No. 11531008), the Natural Science Foundation of Shandong Province (Grant No. ZR2015AM016) and the National Natural Science Foundation of China (Grant No. 11501324)
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2017 American Mathematical Society

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