Around the Bombieri-Vinogradov theorem
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 \(p\), \(p+2\) not exceeding \(X\). It is stated that the coefficient \(64/17\) could be reduced by incorporating techniques due to \textit{Jingrun Chen} [Sci. Sin. 21, 701--739 (1978; Zbl 0399.10046)]. The paper contains an interesting conjecture \(E(k)\), that (1) holds, for some \(u=u_k>0\), when \(x_j=\sum_{j=mn, m\leq X^u}\lambda_m\tau_k(n)\) for each \(A\) and each integer \(k\geq 1\), uniformly for \(1\leq |a| \leq \log^AX\) and for each sequence \(\lambda_m\) satisfying \(\lambda_m\leq \tau_k(m)\). It is shown that if \(E(k)\) holds for \(k=3,4,5,6\) then it holds for all \(k\), and that this would imply (1) when \(x_n\) is the sequence of primes. The results of this paper improve on the corresponding ones contained in the author's thesis for his Doctorat d'État, by using the estimates of \textit{J.-M. Deshouillers} and \textit{H. Iwaniec} [Invent. Math. 70, 219--288 (1982; Zbl 0502.10021)] for averages of Kloosterman sums in place of the estimates of Weil for the sums themselves.
sequence of integers having all prime factors exceeding z
scientific article; zbMATH DE number 3879023
scientific article