Friedlander, John; Granville, Andrew Limitations to the equi-distribution of primes. I. (English) Zbl 0671.10041 Ann. Math. (2) 129, No. 2, 363-382 (1989). The paper considers the distribution of primes in arithmetic progressions. The Bombieri-Vinogradov theorem shows that, for any fixed integer \(a\neq 0\) and any fixed \(A>0\) one has \[ \sum_{q<Q;(q,a)=1}| \psi (x;q,a)-x/\phi (q)| \ll x(\log x)^{-A} \] with \(Q=x^{1/2}(\log x)^{-B}\), for some \(B=B(A)\). P. D. T. A. Elliott and H. Halberstam [Symp. Math. IV, Roma 1968/69, 59-72 (1970; Zbl 0238.10030)] asked whether \(Q=x(\log x)^{-B}\) might be admissable, and H. L. Montgomery [Topics in multiplicative number theory (Lect. Notes Math. 227) (1971; Zbl 0216.03501)] made the still stronger conjecture that \[ \psi (x;q,a)-x/\phi (q)\quad \ll \quad (x/q)^{1/2+\epsilon} \log x, \] uniformly for \((a,q)=1\) and \(q<x\). The present paper disproves these conjectures, but leaves open the possibility that \(Q=x^{1-\epsilon}\) is admissable. Specifically, it is shown that for any fixed \(B>0\) there are arbitrarily large a and x for which \[ \sum_{q<x(\log x)^{-B};(q,a)=1}| \psi (x;q,a)-x/\phi (q)| \quad \gg \quad x. \] The proof is based on the method of H. Maier [Mich. Math. J. 32, 221-225 (1985; Zbl 0569.10023)], who showed that for fixed \(B>0\) one has \[ \psi (x+\log^ B x)-\psi (x)=\log^ B x+\Omega_{\pm}(\log^ B x). \] This was done by choosing x so that the “Fundamental Lemma” of sieve theory gives a higher (or lower) than average a priori probability for the occurrence of primes in the relevant interval. In the present paper a modification of the method is used to handle short arithmetic progressions. Reviewer: D.R.Heath-Brown Cited in 5 ReviewsCited in 27 Documents MSC: 11N05 Distribution of primes 11N13 Primes in congruence classes 11N35 Sieves Keywords:disproof of Elliott-Halberstam conjecture; sieve methods; distribution of primes in arithmetic progressions; Bombieri-Vinogradov theorem; short arithmetic progressions Citations:Zbl 0238.10030; Zbl 0216.03501; Zbl 0569.10023 × Cite Format Result Cite Review PDF Full Text: DOI Link