A Montgomery-Hooley type theorem for prime \(k\)-tuplets
type: Item
instance of: scholarly article
title: A Montgomery-Hooley type theorem for prime \(k\)-tuplets
zbMATH Open document ID: 0869.11069
DOI: 10.1007/BF01874284
published in: Acta Mathematica Hungarica
publication date: 1997-07-16T00:00:00Z
review text: This paper proves an average mean square estimate for prime \(k\)-tuples in long arithmetic progressions. This result is a generalization of the Barban-Davenport-Halberstam theorem for primes in arithmetic progressions. Given \(2k\) integers \(a_j\) and \(b_j\), \(0\leq j\leq k-1\), then the \(k\)-tuplet \((a_0n+b_0, a_1n+b_1, \cdots , a_{k-1}n+b_{k-1})\) is called a prime \(k\)-tuplet for a given \(n\) if each component is prime. The case \(k=2\), \(a_0=a_1=1\), \(b_0=0\), \(b_1=2\) give the familiar twin primes. Letting \({\mathbf b}= (b_0, b_1, \cdots, b_{k-1})\), we count the prime \(k\)-tuplets in the arithmetic progression \(a(\text{mod } q)\) with the function \[ \psi(x; {\mathbf b}, a,q) = \sum_{\scriptstyle n \in N({\mathbf b}) \atop \scriptstyle n \equiv a(\text{mod } q)} \prod_{j=0}^{k-1} \Lambda( a_jn+b_j) \] where \(N({\mathbf b}) = \{t : 1\leq a_jt+b_j\leq x\) for all \(0\leq j\leq k-1 \} \), and \(\Lambda\) is the von Mangoldt function. It has been conjectured that \( \psi(x; {\mathbf b}, a,q) \sim \sigma({\mathbf b}; a,q) |N({\mathbf b})|\) when \( \sigma({\mathbf b}; a,q) >0\), where \(\sigma({\mathbf b}; a,q)\) is the singular series for this problem whose definition is to complicated to give here, and \(|N({\mathbf b})|\) is the length of the interval \(N({\mathbf b})\). Let \(Z=Z(x) = \{{\mathbf b} : |N({\mathbf b})|\neq 0\}\). Now define \[ E(x,Q) = \sum_{q\leq Q}\sum_{a=1}^q\sum_{{\mathbf b}\in Z}\left|\psi(x; {\mathbf b}, a,q) - \sigma({\mathbf b}; a,q) |N({\mathbf b})|\right|^2. \] The main result proved is that for \(Q\leq x/a_*\), where \(a_* = \max_{0\leq j\leq k-1} |a_j|\), \[ \begin{align*}{ E(x,Q) = {1\over \phi(|a_0|)}x^kQ(\log x-1)^k-x^kQ& \sum_{m=0}^k\beta_m (\log{x\over Qa_*})^m \cr &+O(x^{k- {1\over k+2}}Q^{k+3\over k+2}+ {x^{k+1}\over (\log x)^a}),\cr}\end{align*} \] where the numbers \(\beta_m\) depend only on the \(a_0, a_1, \cdots, a_{k-1}\), and further \(\beta_k = 1/(k! \phi(|a_0|))\). A slightly different asymptotic formula …
Mathematics Subject Classification ID: 11N13
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 755501
zbMATH Keywords: average mean square estimate
zbMATH Keywords: prime k-tuples in long arithmetic progressions
zbMATH Keywords: Barban-Davenport-Halberstam theorem
zbMATH Keywords: von Mangoldt function
zbMATH Keywords: asymptotic formula
author: Koichi Kawada
reviewed by: Daniel Alan Goldston
MaRDI profile type: Publication
cites work: THE 'LARGE SIEVE' METHOD AND ITS APPLICATIONS IN THE THEORY OF NUMBERS
cites work: A Heuristic Asymptotic Formula Concerning the Distribution of Prime Numbers
cites work: Primes in arithmetic progressions
cites work: The large sieve
cites work: On the Barban-Davenport-Halberstam theorem. I.
cites work: The prime \(k\)-tuplets in arithmetic progressions
cites work: Q3289922
full work available at URL: https://doi.org/10.1007/bf01874284
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description: scientific article; zbMATH DE number 755501