Trigonometric sums over primes II

We write e ( x ) for e 2πix , ∥x∥ for the distance of x from the nearest integer and use A ≫ B to mean |A|< c | B |, where c is a positive constant depending at most on k and e . The letter p always denotes a prime number; P 2 represents a number with precisely two prime factors. We continue the investigations started in [6] and will make many references to the analysis there. Here we prove the following theorems.