Conditional bounds for small prime solutions of linear equations
type: Item
instance of: scholarly article
title: Conditional bounds for small prime solutions of linear equations
zbMATH Open document ID: 0753.11033
DOI: 10.1007/BF02567674
author: Kai-Man Tsang
author: K. K. Choi
author: Ming-Chit Liu
published in: Manuscripta Mathematica
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/155699
review text: Let \(a_ 1,a_ 2,a_ 3,b\) be integers satisfying \(b\equiv a_ 1+a_ 2+a_ 3\pmod 2\) and \((a_ 1,a_ 2,a_ 3)=(b,a_ i,a_ j)=1\) for \(1\leq i<j\leq 3\). In earlier papers [J. Reine Angew. Math. 399, 109-136 (1989; Zbl 0667.10030) and Monatsh. Math. 111, 147-169 (1991; Zbl 0719.11064)] the second and third author considered solutions to the equation \[ a_ 1p_ 1+a_ 2p_ 2+a_ 3p_ 3=b \tag{*} \] in primes \(p_ 1,p_ 2,p_ 3\). They proved that there are constants \(A_ 1,A_ 2\) such that (i) if \(a_ 1,a_ 2,a_ 3>0\) and \(b\geq (3\max\{a_ 1,a_ 2,a_ 3\})^{A_ 1}\), then \((*)\) is soluble; (ii) if \(a_ 1,a_ 2,a_ 3\) are not of the same sign, then \((*)\) has solutions satisfying \[ \max\{p_ 1,p_ 2,p_ 3\}\leq 3| b|+(3\max\{| a_ 1|,| a_ 2|,| a_ 3|\})^{A_ 2}. \] Observe that (i) includes the classical three primes theorem of Vinogradov, while (ii) can be applied to show that if \(q,\ell\) are coprime, then the least prime \(p\equiv\ell\pmod q\) satisfies \(p\ll q^{A_ 2}\). An estimate on \(A_ 2\) will therefore give an estimate for Linnik's constant, and the first author has shown that \(A_ 1,A_ 2<4191\). In the present paper the authors replace (i) and (ii) by very much more precise estimates established under the assumption of the Generalised Riemann Hypothesis. In particular, the conditional results give \(A_ 1<5+\varepsilon\), \(A_ 2<4+\varepsilon\) for any \(\varepsilon>0\), and also, for the least prime in arithmetic progression problem, the bound \(p\ll q^ 2\log^{10}q\), which is only slightly weaker than the well known bound of \(q^ 2\log^ 2q\) obtainable directly under GRH. Appropriate generating functions have to be introduced so that the problem is brought under the circle method and the relevance of GRH. The new result is based heavily on the ingenious evaluation of a certain generalised Gauss sum.
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 60075
zbMATH Keywords: small prime solutions of linear equations
zbMATH Keywords: least prime in arithmetic progression
zbMATH Keywords: Linnik's constant
zbMATH Keywords: Generalised Riemann Hypothesis
zbMATH Keywords: generalised Gauss sum
reviewed by: Peter Shiu
MaRDI profile type: Publication
cites work: On some diophantine inequalities involving primes.
cites work: Q3819148
cites work: Small prime solutions of some additive equations
Recommended article: An improved bound for prime solutions of some ternary equations
Recommended article: Additive problems involving primes of special type
Recommended article: Q4208318
Recommended article: Q4731258
Recommended article: Q4325423
Recommended article: A bound for prime solutions of some ternary equations
Recommended article: Q4008292
Recommended article: Q4249709
Recommended article: An additive problem with primes in arithmetic progressions
Recommended article: Q4309273
description: scientific article; zbMATH DE number 60075