An estimate of Gaussian sums

S. B. Stechkin

Matematičeskie zametki, Tome 17 (1975) no. 4, pp. 579-588.

Résumé

Suppose $n\in N$, $n\ge3$, $a\in Z$, $q\in N$, $(a,q)=1$. It is shown that for the Gaussian sums $$ S_n(a,q)=\sum_{k=0}^{q-1}e^{2\pi i\frac aqk^n} $$ the following estimate holds uniformly with respect to all parameters: $$ |S_n(a,q)|\le\exp\{C(n\varphi(n))^2\}q^{1-1/n}, $$ where $C$ is a positive absolute constant and $\varphi(n)$ is Euler's function.