On exponential sums over primes in arithmetic progressions

Abstract

with a wide uniformity in real a, where A is the von Mangoldt function and, for real 6, e{6) = exp(2niO). By using a combinatrial identity, R. C. Vaughan presented an elegant simple argument on it, see [2], for instance. J.-r.Chen's theorem on the binary Goldbach problem is built upon the linear sieve and the mean prime number theorem, vide [5]. According to H. Iwaniec [6], the Rosser's weight of the linear sieve has the property. An arithmetic function k is called well-factorable of level D, if for any D\,Di > 1, D = D\Dj, there exist two functions k\and ki supporting in (0,Z>i] and (0,Z>2] respectively such that |^i| < 1, \ki\< 1 and k = k\* kj. Also the mean prime number theorem has been surprisingly developed by E. Fouvry and H. Iwaniec [4],E. Fouvry [3] and E. Bombieri, J.-B. Friedlander and H. Iwaniec [1].In [1] they established a non-trivial bound of the averaging sum