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On the Brun-Titchmarsh theorem on average. (English) Zbl 0548.10026
Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 319-333 (1984).
[For the entire collection see Zbl 0541.00002.]
The number \(\pi\) (x;q,a) of primes \(p\leq x\), \(p\equiv a mod q\), is discussed. A proof is given of the following theorem. Let \(x\geq 2\), \(\epsilon >0\), \(1\leq| a|\leq x^{\epsilon}\) and \(x^{1/2}\leq Q\leq x^{1-\epsilon}\). Then for almost all q in [Q,2Q], \((q,a)=1\), we have \[ \pi (x;q,a)\leq (4/3+\epsilon c_ 1)x/\phi (q)\log (x/q) \] with the number of exceptions less than \(Qx^{-\epsilon c_ 2}\), provided \(x\geq x_ 0(\epsilon)\). Corollary. For infinitely many primes p the largest prime factor of \(p+a\) exceeds \(p^{\vartheta -\epsilon}\), where \(\vartheta =1-{1\over2}e^{-3/8}\). These are improvements of results of the second author in J. Math. Soc. Japan 34, 95-123 (1982; Zbl 0486.10033), obtained with the use of a result from the authors’ paper [Invent. Math. 70, 219-288 (1982; Zbl 0502.10021)].
Reviewer: R.W.Bruggeman

MSC:
11N13 Primes in congruence classes
11N35 Sieves
11N05 Distribution of primes