A Bombieri-Vinogradov type exponential sum result with applications
type: Item
instance of: scholarly article
title: A Bombieri-Vinogradov type exponential sum result with applications
zbMATH Open document ID: 1179.11025
author: Kaisa Matomäki
published in: Journal of Number Theory
publication date: 2009-07-06T00:00:00Z
review text: In this paper the exponential sums \[ S_d=\sum\limits_{n\sim x\atop{n\equiv c\pmod d}}\Lambda (n)e(\alpha n) \] are considered. Here \(\Lambda (n)\) is von Mangoldt function, \(x\) is a large positive number, \(c\neq 0\) is a integer, and \(\alpha\) is a real number with a rational approximation \(a/q\) satisfying \[ \bigg | \alpha -\frac{a}{q} \bigg |<\frac{1}{q^2},\;\;(a,q)=1,\;q\geq 1\,. \] For well-factorable function \(\lambda\) of level \(D\) the author prove that \[ \sum\limits_{d\sim D\atop{(d,c)=1}}\lambda_d \sum\limits_{k\sim K}c_k \sum\limits_{n\sim x\atop{n\equiv c\pmod d}}\Lambda (n)e(\alpha nk) \ll\tag{1} \] \[ K(\log x)^Cx^{3/4+\eta} \bigg( \frac{x}{q}+\frac{q}{K}+D^2+x^{7/9+4\eta}+ \min\bigg\{ d^{4+20\eta}, \frac{x}{D} \bigg\} \bigg)^{1/4-\eta}\,, \] where \(\eta>0\) and \(c_k\ll \tau (k)^C\) for some \(C>0\). This estimate improves the previous result of \textit{H. Mikawa} [Tsukuba J. Math. 24, No. 2, 351--360 (2000; Zbl 1002.11066)]. As a corollary a Bombieri-Vinogradov type result for sums \(S_d\) is obtained. Using (1) the author shows that for any irrational \(\alpha\), real \(\beta\) and some \(\theta>0\) there are infinitely many primes \(p\) such that \(p+2\) has at most two prime factors and \(\|\alpha p+\beta\|<p^{-\theta}\).
reviewed by: Tatyana L. Todorova
Mathematics Subject Classification ID: 11L20
Mathematics Subject Classification ID: 11N13
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 5575285
zbMATH Keywords: exponential sums
zbMATH Keywords: primes in arithmetic progressions
zbMATH Keywords: almost primes
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.01.010
OpenAlex ID: W2022383072
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DOI: 10.1016/J.JNT.2009.01.010
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description: scientific article; zbMATH DE number 5575285