Moments for primes in arithmetic progressions, I

The second momentcenter(sigma)q(more than)Q(sigma)a=1q((psi)(x;q,a)(rho)(x;q,a))^2is investigated with the novel approximation center (rho)(x;q,a)=(sigma)n(more than)xn(identical)a(mod q)FR(n),where center FR(n)= (sigma)r(more than)R(mu)(r)(theta)(r)(sigma)b=1(b,r)=1re(bn/r),and it is shown that when R(more than)logA(kappla), this leads to more precise conclusions than in the classical Montgomery-Hooley case.