On the Vinogradov-Bombieri theorem. (Russian) Zbl 0588.10043

In the usual notation, the Bombieri-Vinogradov theorem states that \[ \sum_{q\leq Q}\max_{(a,q)=1}\max_{y\le x}| \psi (y,q,a)-y/\varphi (q)| \ll x(\log x)^{-A} \tag{*} \] holds with \(Q=\sqrt{x}(\log x)^{-B}\), for any fixed \(A\ge 0\) with a suitable constant \(B=B(A)\). Improving on recent results of R. C. Vaughan and B. V. Levin, the author shows that one may take \(B=A+11/8\) in (*). His result constitutes the currently sharpest form of (*). Attention should be drawn here to recent work of E. Bombieri, J. B. Friedlander and H. Iwaniec (see the preceding review) who showed that \(Q\) can be taken substantially larger than \(\sqrt{x}\) in (*), provided the maximum over \((a,q)=1\) is dropped in the sum and only residue classes a mod q with a fixed integer a are considered.


11N13 Primes in congruence classes


Zbl 0588.10042