ON A VARIANCE ASSOCIATED WITH THE DISTRIBUTION OF PRIMES IN ARITHMETIC PROGRESSIONS

Abstract

As is usual in prime number theory, write $$\psi(x,q,a)=\sum_{{n\le x}\atop{n\equiv a\pmod q}}\Lambda(n).$$ 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.