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Let
$$
\text{Ex}(q, a) := \sum_{n \leq x \atop n \equiv a \,(q)} \Lambda(n) - \frac{x}{\varphi(q)}.
$$
A result of Hooley \cite{Hooley1998} says that for $Q$ close to $x$ (in the Barban-Davenport-Halberstam sense)
$$
\sum_{q \leq Q} \sum_{a=1}^{q}{'}\; \text{Ex}(q, a)^3 \ll Q^2 \left( \frac{x}{Q} \right)^{3/2} e^{-c \sqrt{\log x / Q}}
$$
and recently we proved that the error is in fact
$$
Q^2 \left( \frac{x}{Q} \right)^{1+\epsilon}.
$$
We discuss the relevance of the result and the main idea in the proof.

[1] C. Hooley, On the Barban-Davenport-Halberstam theorem VIII, Journal für die reine und angewandte Mathematik, 499 (1998).