Moments for primes in arithmetic progressions, II

The third moment center(sigma)z(more than)0(sigma)a=1q((psi)(x;q,a)(rho)(x;q,a))^3 is investigated with the novel approximation center(rho)(x;q,a)=(sigma)n(more than)xn(identical)a(mod q)FR(n), where centerFR(n)=(sigma)r(more than)R(M=mu)(r)(theta)(r)(sigma)b=1(b,r) =1re(bn/r), and it is shown that when R(more than)logA(kappa), this leads to more precise conclusions than those obtained by Hooley in the classical case.