Distribution in the mean of arithmetic functions in short intervals in progressions

It is shown that arithmetic functions of a certain class including, in particular, the functions $\Lambda(n)$, $\mu(n)$, and $\tau_r(n)$, on the intervals $x<n\leqslant x+y$, $y>x^{7/12}$, are uniformly distributed in progressions. The result for $\Lambda(n)$ is as follows. Let
$$ \delta(Q,x,y)=\sum_{k\leqslant Q}\max_{(a,k)=1}\max_{\frac x2\leqslant N\leqslant x}\max_{h\leqslant y}\Bigg|\sum_{\substack{N<n\leqslant N+h\\n\equiv a (\operatorname{mod}k)}}\Lambda(n)-\frac h{\varphi(k)}\Bigg|. $$
Then for $x^{3/5}(\log x)^{2(A+64)+1}\leqslant y\leqslant x$ and $Q=yx^{-1/2}(\log x)^{-(A+64)}$ we have $\delta(Q,x,y)\ll y\log^{-A}x$. If $x^{7/12}<y\leqslant x$ then this estimate holds, but with $Q=yx^{-11/20-\delta}$, $\delta>0$.
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