On Averages of Exponential Sums over Primes

In this paper we shall be concerned with obtaining approximations to and estimates for the sum $${{\text{S}}_{\text{N}}}(\alpha ){\text{ = }}\sum\limits_{{\text{n}} \leqslant {\text{N}}} {{\text{e}}({\text{n}}\alpha ) \wedge {\text{(n)}}}$$ where e(x) = exp(2πix), α is real, and Λ(n) is the von Mangoldt function. Although we are unable to establish the naturally conjectured results for this sum, we shall show how the introduction of averaging — in a form likely to occur in applications — can lead to substantial improvements.