On the Sato-Tate conjecture for Kloosterman sums
type: Item
instance of: scholarly article
title: On the Sato-Tate conjecture for Kloosterman sums
zbMATH Open document ID: 0844.11055
DOI: 10.1007/BF01884290
published in: Inventiones Mathematicae
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/144291
review text: Let \(S(a, b; q)\) be the usual Kloosterman sum, and let \(S(1, m; p)= 2\sqrt {p} \cos \vartheta_{p,m}\) with \(0\leq \vartheta_{p,m}\leq \pi\). The Sato-Tate conjecture is that, for fixed \(m\neq 0\), the angles \(\vartheta_{p,m}\) have a distribution function \({2\over \pi} \sin^2 \vartheta\) as the prime \(p\) varies. Here it is shown that, for each prime \(p\in [x,2x ]\) the angles \(\vartheta_{p,q}\) \((0<q< p)\) and \(\vartheta_{p, mq^{-2}}\) \((q\in [x,2x ])\) follow the Sato-Tate distribution. As corollaries one finds that a positive proportion of pairs \(p,q\leq x\) of primes have \(|S(1, m; pq)|\geq (0.16) 4\sqrt {pq}\), and for fixed \(\varepsilon>0\), a positive proportion have \(|S(1, m; pq)|\leq \varepsilon \sqrt {pq}\). The constant \(0.16\) arises because the Sato-Tate distribution accords a probability greater than \({1\over 2}\) to the event \(\cos \vartheta\geq 0.4\). The proofs use estimates for multiple exponential sums, for which a certain amount of algebraic geometry is required. They also depend on a sieve identity (of Buchstab type) to handle the summation over primes.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11N36
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 784019
zbMATH Keywords: almost-primes
zbMATH Keywords: sieve identity of Buchstab type
zbMATH Keywords: Kloosterman sum
zbMATH Keywords: Sato-Tate conjecture
zbMATH Keywords: distribution function
zbMATH Keywords: Sato-Tate distribution
zbMATH Keywords: estimates for multiple exponential sums
Wikidata QID: Q122984466
author: Philippe Michel
MaRDI profile type: Publication
cites work: On a theorem of Bombieri–Vinogradov type
OpenAlex ID: W3136460466
description: scientific article; zbMATH DE number 784019