Moments for primes in arithmetic progressions, II

The third moment \[ \sum\limits_{z \leqslant 0} {\sum\limits_{a = 1}^q {(\psi (x;q,a) - \rho (x;q,a))^3 } } \] ¶ is investigated with the novel approximation \[ \rho (x;q,a) = \sum\limits_\substack{ n \leqslant x \\ n \equiv a({\text{mod}}\;q) } {F_R (n),} \] ¶ where \[ F_R (n) = \sum\limits_{r \leqslant R} {\frac{{\mu (r)}} {{\phi (r)}}\sum\limits_\substack{ b = 1 \\ (b,r) = 1 } ^r {e(bn/r),} } \] ¶ and it is shown that when R\leq \log \sp Ax , this leads to more precise conclusions than those obtained by Hooley in the classical case.