Sums of \(k\)-th powers and the Whittaker-Fourier coefficients of automorphic forms
type: Item
instance of: scholarly article
title: Sums of \(k\)-th powers and the Whittaker-Fourier coefficients of automorphic forms
zbMATH Open document ID: 1487.11045
published in: The Ramanujan Journal
publication date: 2021-08-19T00:00:00Z
review text: The author begins this paper by briefly reviewing the literature on shifted convolution sums. Letting \(\tau_2(n) := \sum_{d_1 d_2 = n} 1\) is the number of divisors of \(n\), Luo reminds the reader that \[ \sum_{n \leq x} \tau_2(n) \tau_2(n+1) \sim \frac{6}{\pi^2} x (\log x)^2 \] as well as some generalizations and strengthenings of this result. He then defines \[ r_s(n) := \# \{(x_1, \dots, x_s) \in \mathbb{Z}^s \ : \ n = \sum_{i = 1}^s x_i^2 \} \] and then recalls that \[ \sum_{n \leq x} r_s(n) a_f(n + 1) = O(x^{s/2-\delta_s}), \] where \(\{a_f(n)\}_n\) are the normalized coefficients of a holomorphic cusp form, or of a Maass cusp form, for \(\Gamma_0(N)\). Let \[ r_{s, k}(n) := \# \{(x_1, \dots, x_s) \in \mathbb{Z}_{>0}^s \ : \ n = \sum_{i = 1}^s x_i^2 \}. \] The author then proceeds to the meat of the paper, which includes several results in the same spirit as the last asymptotic bound in the first paragraph. All of his results are proven by the circle method. Let \(\phi(x)\) be a smooth function supported on \([\frac{1}{2}, 1]\) Luo's Theorem 1 says that for all \(s\) sufficiently large relative to \(k\), \[ \sum_{n \geq 1} a_f(n+1) r_{s, k} \phi(n/x) = O(x^{s/k - \delta} \] for some \(\delta > 0\), and his Theorem 2 gives more precise results in the same vein when \(s \in {4, 5}\) and \(k = s - 1\). Luo's Theorem 1 and Theorem 3 give similar estimates for \(\sum_{n \geq 1} a_\pi(1, 1, 1, \dots, n+1) r_{s, k} \phi(n/x)\), where \(\pi\) is an even Maass automorphic cusp form on \(\mathrm{GL}(m, \mathbb{Z})\). We advise the reader of a small error in the paper, albeit one that does not affect the substance of Luo's results: Luo incorrectly asserts that \(r_{s, 2}(n) = r_s(n)\); however, each \((x_1, \dots, x_s)\) in the defining set for \(r_{s, 2}(n)\) yields \(2^s\) elements \((\pm x_1, \dots, \pm x_s)\) in the defining set for \(r_s(n)\), and there may be additional elements in the defining set for \(r_s(n)\) which have some 0 components.
reviewed by: Grant Molnar
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11P05
zbMATH DE Number: 7383310
zbMATH Keywords: automorphic forms
zbMATH Keywords: Whittaker-Fourier coefficients
zbMATH Keywords: Vinogradov mean value
zbMATH Keywords: circle method
author: Wenzhi Luo
MaRDI profile type: Publication
full work available at URL: s11139-020-00332-4
OpenAlex ID: W3127009294
cites work: On the Vinogradov mean value
cites work: Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three
cites work: Q3767413
cites work: An Additive Divisor Problem
cites work: Kloosterman sums and Fourier coefficients of cusp forms
cites work: Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method
cites work: Automorphic Forms and L-Functions for the GroupGL(n, R)
cites work: Q4350164
cites work: Q4790111
cites work: Shifted convolution sums for higher rank groups
cites work: Q5719475
cites work: Shifted convolution of cusp-forms with \(\theta \)-series
cites work: Cancellation in additively twisted sums on GL(n)
cites work: Shifted convolution sums for \(\mathrm{GL}(3)\times\mathrm{GL}(2)\)
cites work: Hyper-Kloosterman sums of different moduli and their applications to automorphic forms for \(\mathrm{SL}_m(\mathbb{Z})\)
cites work: Shifted Convolution Sums of GL3 Cusp Forms with θ-Series
cites work: The Hardy-Littlewood method.
cites work: The cubic case of the main conjecture in Vinogradov's mean value theorem
cites work: The asymptotic formula in Waring's problem
DOI: 10.1007/S11139-020-00332-4
description: scientific article; zbMATH DE number 7383310