A Bombieri-type mean-value theorem concerning exponential sums over primes (Q1913932)

By analytical means, in particular zero density results for \(L\)-functions, the authors prove the following Bombieri-Vinogradov type mean value theorem for the prime number sum \(S(x, \alpha) = \sum_{n \leq x} \Lambda (n) e(n \alpha)\). Let \(A > 0\), \(B = A + 13\), \(1 \leq Q \leq x^{1/3} (\ln x)^{- 2B/3}\), \(\Theta = \min (Q^{-3} (\ln x)^{- 2B}\), \((\ln x)^{- 8B})\). Then \[ \sum_{q \leq Q} \;\max_{y \leq x} \;\max_{|\lambda |\leq \Theta} \;\max_{(a,q) = 1} \left |S \left( y, {a \over q} + \lambda \right) - {\mu (q) \over \varphi (q)} \sum_{n \leq y} e(n \lambda) \right |\ll {x \over (\ln x)^A}. \] This is an improvement of a similar result of the reviewer [Acta Math. Hung. 61, 241-258 (1993; Zbl 0790.11075)].