Abstract
For any fixed $N > 0$ we show that there exist arbitrarily large values of $a$ and $x$ for which
\[
\mathop{\sum_{q < x/\log^N x}}_{(q,a)=1} \vert\psi(x;q,a) - x/\phi(q)\vert \gg_N x.
\]
This disproves conjectures of Elliott and Halberstam, and of Montgomery. We also establish a number of related results.
