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 ? ?/q? of rank r: ∑_(a∈A) χ (n) _r q~(1/2)(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) ∈ ?[x] is a polynomial of degree 2 ≤ d < q.
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机译：令χ（mod q）为原始Dirichlet字符。在本文中，我们证明了在所有适当的广义算术级数A？上字符和∑_（a∈A）χ（a）的一致上界。 / q等级r的平方：∑_（a∈A）χ（n） _ r q〜（1/2）（log q）〜r。这概括了Pólya和Vinogradov的经典结果。我们的方法还适用于为多项式指数和∑〜（n∈Aeq）（h（n））（q素数）给出统一的上限，其中h（x）∈？[x]是2≤的多项式d 展开▼
