Let χ (mod q) be a primitive Dirichlet character. In this paper, we prove a uniform upper bound of the character sum ∑a∈A χ(a) over all proper generalized arithmetic progressions A ⊂ ℤ/qℤ of rank r: ∑a∈A χ (n) <<r q1/2(log q)r. This generalizes the classical result by Pólya and Vinogradov. Our method also applies to give a uniform upper bound for the polynomial exponential sum ∑ n∈Aeq(h(n)) (q prime), where h(x) ∈ ℤ[x] is a polynomial of degree 2 ≤ d < q.
|Number of pages||10|
|Journal||Bulletin of the London Mathematical Society|
|State||Published - Jun 2013|
ASJC Scopus subject areas
- Mathematics (all)