density hypothesis

For $n\in\mathbb{Z}_{\geq 1}$ let $\Lambda(n)=\log p$ if $n=p^{k}$ for some prime $p$ and integer $k\geq 1$, and $\Lambda(n)=0$ otherwise, and let $\varphi$ denote the Euler totient function. For real $y\geq 0$ and integers $q\geq 1$, $a$, put

$$\psi (y;q,a)=\sum _{n\leq y \atop n\equiv a{\bmod {q}}}\Lambda (n).$$

Let $A>0$. Then there is a constant $C=C(A)$ such that for all real $x\geq 2$ and all real $Q$ with

$$x^{1/2}\log ^{-A}x\leq Q\leq x^{1/2},$$

one has

$$\sum _{q\leq Q}\ \max _{y\leq x}\ \max _{1\leq a\leq q \atop (a,q)=1}\left|\psi (y;q,a)-{y \over \varphi (q)}\right|\ \leq \ C\,x^{1/2}Q(\log x)^{5},$$

the outer sum being over integers $q$ with $1\leq q\leq Q$.