  We prove a new mean-value theorem for Dirichlet polynomials with coefficients
given by the von Mangoldt function. We then use our theorem to derive new
estimates for certain exponential sums over primes. The latter have
applications to additive problems with prime variables. In particular, we are
able to improve on a recent result of J.Y. Liu and K.M. Tsang on the size of
the solutions of linear equations with prime variables.
