Sums of \(k\)-th powers and the Whittaker-Fourier coefficients of automorphic forms
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.
scientific article; zbMATH DE number 7383310
scientific article