Sums of Fourier coefficients of cusp forms of level $D$ twisted by exponential functions over arithmetic progressions
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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 $(l,N)=1$, $c|N$, $D_2=D/(c, D)$ 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 $\alpha \in \mathbb {R}$.References
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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
- MR Author ID: 1152545
- Email: zhmengsdu@hotmail.com
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3761-3774
- MSC (2010): Primary 11L07, 11F30; Secondary 11B25
- DOI: https://doi.org/10.1090/proc/13536
- MathSciNet review: 3665031