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Generalized Bombieri-Vinogradov theorems in short intervals. (Théorèmes généralisés de Bombieri-Vinogradov dans les petits intervalles.) (French) Zbl 0801.11038

The usual form of the well-known Bombieri-Vinogradov theorem states that for any fixed \(A>0\) there exists a constant \(B= B(A) >0\) such that the bound \[ \sum_{q\leq Q} \max_{(a,q)=1} \max_{z\leq x} |\psi (z;q,a)- z/\varphi (q)| \ll_ A x(\log x)^{-A} \] holds with \(Q= x^{1/2} (\log x)^{-B}\). Among the many generalizations of this theorem is one by Ding and Pan [C. D. Pan, Recent progress in analytic number theory, Durham 1979, Vol. 1, 275–287 (1981; Zbl 0462.10032)] in which the differences \(\psi(z; q,a)- z/ \varphi(q)\) are replaced by averages of the form \[ \sum_{l\leq x^{1-\varepsilon}} g(l) \Biggl\{ \sum _{\substack{ nl\leq z\\ nl\equiv a\pmod q }} \Lambda(n)- {{z/l} \over {\varphi(q)}} \Biggr\}, \] where \(g(l)\) is an arbitrary weight function satisfying \(g(l)\ll 1\).
In the paper under review, the author proves an analogous generalization of a short interval Bombieri-Vinogradov theorem due to A. Perelli, J. Pintz and S. Salerno [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11, 529–539 (1984; Zbl 0579.10019)]. He applies this result to obtain the following localized version of Chen’s theorem: For any \(\theta\) satisfying \(0.974\leq \theta\leq 1\) and all sufficiently large \(x\), there are \(\gg x^ \theta/ \log^ 2 x\) primes \(p\) in the interval \((x,x+ x^ \theta]\) such that \(p+2\) has at most two prime factors.

MSC:

11N13 Primes in congruence classes
11N25 Distribution of integers with specified multiplicative constraints
11P32 Goldbach-type theorems; other additive questions involving primes
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