  We give an effective version with explicit constants of a mean value theorem
of Vaughan related to the values of \psi(y, \chi), the twisted summatory
function associated to the von Mangoldt function \Lambda and a Dirichlet
character \chi. As a consequence of this result we prove an effective variant
of the Bombieri-Vinogradov theorem with explicit constants. This effective
variant has the potential to provide explicit results in many problems. We give
examples of such results in several number theoretical problems related to
shifted primes.
