Primes in arithmetic progressions to large moduli
This paper may be viewed as an extension of the work of \textit{É. Fouvry} and \textit{H. Iwaniec} [Acta Arith. 42, 197--218 (1983; Zbl 0517.10045)]. The aim is to prove variants of the Bombieri-Vinogradov prime number theorem of the form \[ \sum_{_{\substack{ q\leq Q,\\ (a,q)=1}}}\lambda (q)(\psi (x;q,a)-x/\varphi (q))\ll x(\log x)^{-A}, \tag{*} \] for fixed \(a\neq 0\) and \(A>0\), in which \(Q\) may be larger than \(\sqrt{x}\). Different results are obtained, depending on the form of the weight \(\lambda(q)\). Thus (Theorem 9) if \(\lambda (q)=1\) one may take \(Q=x(\log x)^{-B(A)}\) in (*). This leads to the following form of the Titchmarsh divisor problem: If \(a\neq 0\) and \(A>0\) are given then \[ \sum_{| a| <n\leq x}\Lambda (n) d(n+a)=c_ 1(a) x \log x+c_ 2(a)x+O(x(\log x)^{-A}). \] Another class of weights are the well factorable ones: \(\lambda(q)\) is ``well factorable of level \(Q\)'' if for any \(R>0\) there exist functions \(\alpha(q)\), \(\beta(q)\), supported on \([1,R]\) and \([1,Q/R]\), respectively, and for which \(| \alpha (q)|,\quad | \beta (q)| \leq 1\) and \(\lambda (q)=\sum_{d| q}\alpha (q)\beta (q/d)\). For such weights one may take \(Q=x^{4/7-\varepsilon}\) in (*). Well factorable weights arise in the linear sieve \textit{[H. Iwaniec}, Acta Arith. 37, 307--320 (1980; Zbl 0444.10038)] and consequently one gets a new bound \[ \pi_ 2(x)\leq (7/2+o(1)) Bx(\log x)^{-2},\quad B=2\prod_{p>2}(1- 1/(p-1)^ 2) \] for the number of twin primes up to \(x\). The proofs use averages of incomplete Kloosterman sums due to \textit{J.-M. Deshouillers} and \textit{H. Iwaniec} [Invent. Math. 70, 219--288 (1982; Zbl 0502.10021)]. From these are derived a number of bounds of the type given by (*), but with the function \(\Lambda(n)\) which occurs in \(\psi(x;q,a)\) replaced by a convolution of functions supported on shorter ranges. Such estimates appear to have other possible applications, and a conjecture is made as to the best possible inequality one might hope for.
scientific article; zbMATH DE number 3943938
scientific article