Exponential sums with Fourier coefficients of automorphic forms

2017 ◽  
Vol 289 (1-2) ◽  
pp. 267-278 ◽  
Author(s):  
Guangshi Lü
Keyword(s):  
Fourier Coefficients ◽  
Exponential Sums ◽  
Automorphic Forms

On exponential sums for coefficients of general L-functions

2021 ◽  
pp. 1-22
Author(s):  
Fei Hou
Keyword(s):  
Functional Equation ◽  
General Formula ◽  
Fourier Coefficients ◽  
Exponential Sums ◽  
Exponential Functions ◽  
Maass Forms ◽  
Maass Form ◽  
Rapid Decay

We investigate the order of exponential sums involving the coefficients of general [Formula: see text]-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for [Formula: see text], Sci. China Math. 58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math. 57 (2017) 185–204].

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 Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients

2021 ◽  
pp. 1-35
Author(s):  
Nadiia Derevianko ◽  
Gerlind Plonka
Keyword(s):  
Rational Approximation ◽  
Fourier Coefficients ◽  
Exponential Sums ◽  
Finite Interval ◽  
Large Set ◽  
Recovery Method ◽  
Exact Reconstruction ◽  
Finite Set ◽  
Number Formula ◽  
Numerical Approaches

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [Formula: see text], where the frequency parameters [Formula: see text] are pairwise distinct. In order to reconstruct [Formula: see text] we employ a finite set of classical Fourier coefficients of [Formula: see text] with regard to a finite interval [Formula: see text] with [Formula: see text]. For our method, [Formula: see text] Fourier coefficients [Formula: see text] are sufficient to recover all parameters of [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The recovery is based on the observation that for [Formula: see text] the terms of [Formula: see text] possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of [Formula: see text] Fourier coefficients of [Formula: see text] is available (i.e. [Formula: see text]), then our recovery method automatically detects the number [Formula: see text] of terms of [Formula: see text], the multiplicities [Formula: see text] for [Formula: see text], as well as all parameters [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], determining [Formula: see text]. Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

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