Abstract
<abstract><p>Let $ F $ be a normlized Hecke-Maaß form for the congruent subgroup $ \Gamma_0(N) $ with trivial nebentypus. In this paper, we study the problem of the level aspect estimates for the exponential sum</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathscr{L}_F(\alpha) = \sum\limits_{n\le X} A_F(n, 1)e(n \alpha). $\end{document} </tex-math></disp-formula></p>
<p>As a result, we present an explicit non-trivial bound for the sum $ \mathscr{L}_F(\alpha) $ in the case of $ N = P $. In addition, we investigate the magnitude for the non-linear exponential sums with the level being explicitly determined from the sup-norm's point of view as well.</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference15 articles.
1. A. Corbett, Voronoĭ summation for $GL_n$: collusion between level and modulus, Am. J. Math., 143 (2021), 1361–1395. http://dx.doi.org/10.1353/ajm.2021.0034
2. D. Godber, Additive twists of Fourier coefficients of modular forms, J. Number Theory, 133 (2013), 83–104. http://dx.doi.org/10.1016/j.jnt.2012.07.010
3. G. Harcos, P. Michel, The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points, Ⅱ, Invent. Math., 163 (2006), 581–655. http://dx.doi.org/10.1007/s00222-005-0468-6
4. G. Hardy, J. Littlewood, Some problems of Diophantine approximation, Acta Math., 37 (1914), 193–239. http://dx.doi.org/10.1007/BF02401834
5. F. Hou, A explicit Voronoï formula for $SL_3(\mathbb{R})$ newforms underlying the symmetric square lifts in the level aspect, Pre-print, 2021.