Bounds for exponential sums modulo \(p^2\)
type: Item
instance of: scholarly article
title: Bounds for exponential sums modulo \(p^2\)
zbMATH Open document ID: 1204.11133
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-10-28T00:00:00Z
review text: 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 \(p^{7/10}<t<p\). \[ S(G)\ll\begin{cases} (p^7t^{26})^{1/36}\quad\text{if}\;p^{7/10}<t<p^{3/4}(\log p)^{-1},\\ p^{1/9}t^{5/6}(\log p)^{1/9}\quad\text{if}\;p^{3/4}(\log p)^{-1}<t<p^{7/9},\\ (p^5t^{17})^{1/24}(\log p)^{1/12}\quad\text{if}\;p^{7/9}<t<p^{4/5}, \\ p^{3/8}t^{1/2}\quad\text{if}\;p^{4/5}<t<p. \end{cases} \]
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 5358119
author: Yuri V. Malykhin
MaRDI profile type: Publication
full work available at URL: s10958-007-0385-x
OpenAlex ID: W2073641830
cites work: Exponential sum estimates over subgroups of \(\mathbb Z^*_q\), \(q\) arbitrary
cites work: Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order.
cites work: Q4885320
cites work: New bounds for Gauss sums derived from KTH powers, and for Heilbronn's exponential sum
cites work: Fractional parts of functions of a special form
cites work: Kloosterman double sums
cites work: Q5296656
cites work: Q4264395
cites work: Q4003239
cites work: Q5587679
DOI: 10.1007/S10958-007-0385-X
description: scientific article; zbMATH DE number 5358119