Bounds for exponential sums modulo \(p^2\)
In this paper the author considers exponential sums \(S(a,G)=\sum_{x\in G}e_q(ax)\) over subgroups \(G \subset \mathbb Z_q^*\), \(\#G=t\), \(a\in\mathbb Z_q\). Put \(S(G):=\max_{a\in\mathbb Z_q^*} | S(a,G)|\). Using Stepanov's method, he obtains nontrivial bounds for exponential sums \(S(G)\) in the case where \(q\) is a square of a prime number. The case \(t\geq p\) is clear, for then \(S(G)=0\). The following estimates for \(S(G)\) are given in the range \(p^{7/10}<t<p\). \[ S(G)\ll\begin{cases} (p^7t^{26})^{1/36}\quad\text{if}\;p^{7/10}<t<p^{3/4}(\log p)^{-1},\\ p^{1/9}t^{5/6}(\log p)^{1/9}\quad\text{if}\;p^{3/4}(\log p)^{-1}<t<p^{7/9},\\ (p^5t^{17})^{1/24}(\log p)^{1/12}\quad\text{if}\;p^{7/9}<t<p^{4/5}, \\ p^{3/8}t^{1/2}\quad\text{if}\;p^{4/5}<t<p. \end{cases} \]
scientific article; zbMATH DE number 5358119