On Hua's Estimate for Exponential Sums

Pure and mixed exponential sums

“…In order to improve on Hua's estimate, two different approaches have been taken. Smith [19], Loxton and Smith [14], and Loxton and Vaughan [15] considered the factorization of f (x) over the complex plane and in the latter paper it was shown that…”

Multifractal behavior of polynomial Fourier series

“…In the case P (x) = c 0 x k , we have Hua's bound [LS82] that reads τ m ≪ d k−1 (q)q 1/2 (m, q) 1/2 with d k−1 the k − 1 divisor function [Ivi03]. The elementary bound m≍M (m, q) ≪ d(q)M shows that (2.2) and (2.3) are still true with d(q) in the right hand side when introducing (m, q) 1/2 in the summation.…”

On upper bounds of Chalk and Hua for exponential sums

“…Another approach for bounding exponential sums was taken up in the work of Smith [42], Loxton and Smith [23] and Loxton and Vaughan [24], with the latter obtaining the bound…”