On character sums and exponential sums over generalized arithmetic progressions

Author: Shao Xuancheng  

Publisher: Oxford University Press

ISSN: 0024-6093

Source: Bulletin of the London Mathematical Society, Vol.45, Iss.3, 2013-06, pp. : 541-550

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Abstract

Let (mod q) be a primitive Dirichlet character. In this paper, we prove a uniform upper bound of the character sum aa(a) over all proper generalized arithmetic progressions a/q of rank r: This generalizes the classical result by Plya and Vinogradov. Our method also applies to give a uniform upper bound for the polynomial exponential sum naeq(h(n)) (q prime), where h(x)[x] is a polynomial of degree 2d<q.