Sums of Fourier coefficients of cusp forms of level D twisted by exponential functions

Abstract

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χ_D, and let a_g(n) be its n-th Fourier coefficient. We consider the sum and prove that S1 has an asymptotic formula when β = 1/2 and αis close to \pm \mathrm{2}\sqrt{q/D} for positive integer q\le X/\mathrm{4} and X sufficiently large. And when 0<β<1 and α, β fail to meet the above condition, we obtain upper bounds of S₁. We also consider the sum {{\displaystyle S}}_{\mathrm{2}}={\displaystyle {\sum }_{n>\mathrm{0}}{{\displaystyle a}}_g}(n)e({an}^{\beta })\varphi (n/X) with \varphi (x)\in {{\displaystyle C}}_c^{\infty }(\mathrm{0},+\infty ) and prove that S₂ has better upper bounds than S₁ at some special α and β.

Key words： exponential sums; cusp form; Fourier coefficients

References

1 DeligneP. La conjecture de Weil.I. Inst Hautes Études Sci Publ Math, 1974, 43(1): 273–307

2 GradshteynI S, RyzhikI M. Table of Integrals, Series, and Products. 7th ed.New York: Academic Press, 2007

3 HuxleyM N. Area, Lattice Points, and Exponential Sums.London Math Soc Monogr New Ser, Vol 13. New York: Clarendon Press, Oxford Univ Press, 1996

4 IwaniecH, KowalskiE. Analytic Number Theory.Colloq Publications, Vol 53. Providence: Amer Math Soc, 2004

5 IwaniecH, LuoW, SarnakP. Low lying zeros of families of L-functions.Inst Hautes Études Sci Publ Math, 2000, 91: 55–131

6 KimH, SarnakP. Refined estimates towards the Ramanujan and Selberg conjectures (Appendix to: Kim H. Functoriality for the exterior square of GL4 and the symmetric fourth of GL2).J Amer Math Soc, 2003, 16: 139–183

7 KowalskiE, MichelP, VanderkamJ. Rankin-Selberg L-functions in the level aspect.Duke Math J, 2002, 114(1): 123–191

8 LiuKui, RenXiumin. On exponential sums involving fourier coefficients of cusp forms.J Number Theory, 2012, 132(1): 171–181

9 MillerS D, SchmidW. The highly oscillatory behavior of automorphic distributions for SL(2).Lett Math Phys, 2004, 69(1): 265–286

10 PittN J E. On cusp form coefficients in exponential sums.Quart J Math, 2001, 52: 485–497

11 RenXiumin, YeYangbo. Resonance between automorphic forms and exponential functions.Sci China Math, 2010, 53(9): 2463–2472

12 ShahidiF. Best estimates for Fourier coefficients of Maass forms.In: Automorphic Forms and Analytic Number Theory (Montreal, PQ, 1989). Montreal: Univ Montréal, 1990, 135–141

13 SunQingfeng, WuYuanying. Exponential sums involving Maass forms.Front Math China, 2014, 9(6): 1349–1366

14 WeiBin. Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ).Int J Number Theory, 2015, 11(1): 39–49

15 WiltonJ R. A note on Ramanujan’s arithmetical function τ(n).Math Proc Cambridge Philos Soc, 1929, 25: 121–129