Around the Bombieri-Vinogradov theorem
type: Item
instance of: scholarly article
title: Around the Bombieri-Vinogradov theorem
zbMATH Open document ID: 0552.10024
DOI: 10.1007/BF02392198
author: Étienne Fouvry
published in: Acta Mathematica
publication date: 1984-01-01T00:00:00Z
review text: For sequences \((x_n)\) of real numbers write \[ \sum_{n\leq X; (n,k)=1; n\equiv a \bmod q}x_ n=\{1/\varphi(q)\} \sum_{n\leq X, (n,qk)=1} x_n+E_k((x_n),X,q,a), \] \[ S(X,Q,a)=\sum_{q\leq Q; (q,a)=1} | E_1((x_ n),X,q,a)|. \] The author describes circumstances in which one has, for given \(a\), \(A>0\), \(\eta >0,\) \[ S(X,X^{\theta -\eta},a)\ll X \log^{-A}X;\qquad \theta >1/2, \tag{1} \] so that \(\theta\) exceeds the value featuring in the theorem of the title. The situation studied is that when the sequence \((x_n)\) is given as a convolution \[ x_j=\sum_{j=mn; M<m\leq 2M; N<n\leq 2N} \alpha_ m\beta_ n \] of two others. The sequence \((\beta_ n)\) has to be well-distributed with respect to arithmetic progressions in the sense that \[ E_k((\beta_ n),x,q,b)\ll x \tau_2(k) \log^{-B}x \] for each real \(B\) and each integer \(k\) whenever \((b,q)=1\). Here and below \(\tau_{\gamma}(n)\) denotes the number of representations of \(k\) as a product of \(\gamma\) factors. Write \(N=X^{\nu}\). The sequence \((\beta_n)\) has also to be short in the sense that \(\nu <1/6\). When \(\nu \geq \nu_0>0\) the author establishes (1) with a certain \(\theta >1/2\), subject to a further technical condition \(| \alpha_k|, | \beta_k| \leq (\log k \tau_{\gamma}(k))^{\gamma}+1.\) An interesting application of this theorem is to the case when \((x_m)\) is the sequence of integers having all prime factors exceeding \(z\). Then (1) follows uniformly when \(z\leq X^{(1/6)-\nu_0}\) and \(|a| \leq \log^Ax\). Here the value \(1/6\) improves on the value \(1/883\) appearing in the author's earlier work with \textit{H. Iwaniec} [Mathematika 27, 135--152 (1980; Zbl 0469.10027)]. The theorem also has consequences relating to work of the author with \textit{H. Iwaniec} on primes in arithmetic progressions [Acta Arith. 42, 197--218 (1983; Zbl 0517.10045)]. As a corollary this yields a new upper bound \[ 2(64/17+\varepsilon) \prod_{p>2}\{1-1/(p-1)^ 2\} X/\log^ 2X \] for the number of prime twins \(p\),…
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 3879023
zbMATH Keywords: Bombieri-Vinodgradov theorem
zbMATH Keywords: application
zbMATH Keywords: sequence of integers having all prime factors exceeding z
zbMATH Keywords: primes in arithmetic progressions
zbMATH Keywords: upper bound
zbMATH Keywords: number of prime twins
zbMATH Keywords: conjecture E(k)
reviewed by: George Greaves
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02392198
OpenAlex ID: W2317319582
description: scientific article; zbMATH DE number 3879023