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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