On an exponential sum

P. Ding

Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 63-75

Résumé

Let $p$ be a prime number, $n$ be a positive integer, and $f(x) = ax^k+bx$. We put $$ S(f,p^n)=\sum_{x=1}^{p^n}e\biggl(\frac{f(x)}{p^n}\biggr), $$ where $e(t)=\exp(2\pi it)$. This special exponential sum has been widely studied in connection with Waring's problem. We write $n$ in the form $n=Qk+r$, where $0\le r\le k-1$ and $Q\ge 0$. Let $\alpha=\operatorname{ord}_p(k)$, $\beta=\operatorname{ord}_p(k-1)$, and $\theta=\operatorname{ord}_p(b)$. We define $$ \mathcal Q=\begin{cases} \dfrac{\theta-\alpha}{k-1},\text{если }\theta\ge\alpha, \\ 0,\text{иначе}, \end{cases} $$ and $J=[\zeta]$. Moreover, we denote $V=\min(Q,J)$. Improving the preceding result, we establish the theorem. Theorem. Let $k\ge 2$ and $n\ge 2$. If $p>2$, then $$ |S(f,p^n)|\le\begin{cases} p^{\frac{1-V}2}p^{\frac n2}(b,p^n)^{\frac12},\text{if }n\equiv 1\pmod k, \\ (k-1,p-1)p^{-\frac V2}p^{\frac{\min(\alpha,1)}2}p^{\min(\frac\beta2,\frac n2-1)}p^{\frac n2}(b,p^n)^{\frac12}, \text{if }n\not\equiv 1\pmod k. \end{cases} $$ An example showing that this result is best possible is given. Bibliography: 15 titles.