Exponential sums of squares of Fourier coefficients of cusp forms

2020 ◽  
Vol 130 (1) ◽  
Author(s):  
Ratnadeep Acharya
Keyword(s):  
Fourier Coefficients ◽  
Exponential Sums ◽  
Sums Of Squares ◽  
Cusp Forms

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

scholarly journals A relation between Fourier coefficients of holomorphic cusp forms and exponential sums

2009 ◽  
Vol 86 (100) ◽  
pp. 97-105 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen
Keyword(s):  
Fourier Coefficients ◽  
Exponential Sums ◽  
Cusp Forms ◽  
Lehmer Conjecture ◽  
Holomorphic Cusp Forms

We consider certain specific exponential sums related to holomorphic cusp forms, give a reformulation for the Lehmer conjecture and prove that certain exponential sums of Fourier coefficients of holomorphic cusp forms contain information on other similar non-overlapping exponential sums. Also, we prove an Omega result for short sums of Fourier coefficients.

scholarly journals On triple correlations of Fourier coefficients of cusp forms

2018 ◽  
Vol 183 ◽  
pp. 485-492 ◽  
Author(s):  
Guangshi Lü ◽  
Ping Xi
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms
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