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Limitations to the equi-distribution of primes. III. (English) Zbl 0743.11048

[Part II (to appear); Part IV, cf. Proc. R. Soc. Lond., Ser. A 435, 197- 204 (1991; Zbl 0736.11049).]
If one fixes \(a\neq 0\) and \(N>0\) it is shown that the asymptotic formula (*) \(\pi(x;q,a)\sim\pi(x)/\varphi(q)\) \((a,q)=1\) cannot hold uniformly for \(q\leq x(\log x)^{-N}\). Indeed \(\Omega_ \pm\) results are obtained which give not only many “bad” values of \(q\), but also have a certain uniformity in \(a\). This allows one to say that (*) is false for \(\gg_ \varepsilon x^ 2/\exp\exp(\log^ \varepsilon x)\) pairs \(a,q\) with \(q\leq x(\log x)^{-N}\). In contrast the Barban-Davenport-Halberstam theorem shows that there are \(O(x(\log x)^{1-N})\) such exceptional pairs.
The paper extends the authors’ earlier work [Ann. Math., II. Ser. 129, 363-382 (1989; Zbl 0671.10041)] in which it was shown that (*) cannot hold for all \(a\pmod q\) uniformly for \(q\leq x(\log x)^{-N}\). The ideas of the present paper build on those of the earlier one, and are ultimately motivated by H. Maier’s method [Mich. Math. J. 32, 221- 225 (1985; Zbl 0569.10023)] for handling primes in short intervals.

MSC:

11N13 Primes in congruence classes
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References:

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