As is usual in prime number theory, write$$\psi(x,q,a)=\sum_{{n\le x}\atop{n\equiv a\pmodq}}\Lambda(n).$$It is well known that when $q$ is close to $x$ the average value of$$V(x,q)=\sum_{{a=1}\atop{(a,q)=1}}^q\left|\psi(x,q,a)-{x\over{\phi(q)}}\right|^2$$is about $x\log q$, and recently Friedlander and Goldston have shown that if $$U(x,q)=x\log q-x\left(\gamma+\log 2\pi+\sum_{p|q}\frac{\log p}{p-1}\right),$$then the first moment of $V(x,q)-U(x,q)$ is small.In this memoir it is shown that the same is true for all moments. 2000 Mathematics Subject Classification: 11N13.
