Primes in arithmetic progressions on average I

Abstract

Let $E_x(q,a)$ be the error term when counting primes in arithmetic progressions and let $M(Q)={\sum }_{q\le Q}\ddot{I}•(q){\sum }_{a=1}^q{E}_x(q,a{)}^3$. We show that $M(Q)<<Q^3(x/Q{)}^{7/5}$ for large $Q$ close to $x$ (in the usual BDH sense) thereby showing that sign changes in the error give power saving cancellation past the expected $\sqrt{x/q}$ heuristic.

the 7/5 exponent may be pushed down to 1