= Solution
Let $\mathbf1(n)=1$ and $\epsilon(n)=1_{n=1}$, the identity for <Dirichlet convolution>. The elementary identities needed are
$$
\mu*\mathbf1=\epsilon,\qquad\Lambda*\mathbf1=\log,\qquad\mu*\log=\Lambda.
$$
For the first, <prime factorization> gives $\sum_{d\mid n}\mu(d)=\prod_{p\mid n}(1-1)$ when $n>1$, and one when $n=1$. For the second, if $n=\prod p^{a_p}$, then $\sum_{d\mid n}\Lambda(d)=\sum_pa_p\log p=\log n$, the <Von Mangoldt divisor identity>. Convolving the second identity with $\mu$ proves the third.
For $U,V\geq1$, let the subscripts $\leq U$, $>U$, $\leq V$, $>V$ denote truncations. The <Vaughan identity> is
$$
\boxed{\Lambda=\Lambda_{\leq V}+\mu_{\leq U}*\log-\mu_{\leq U}*\mathbf1*\Lambda_{\leq V}+\mu_{>U}*\mathbf1*\Lambda_{>V}.}
$$
Its pointwise form is
$$
\Lambda(n)=\Lambda(n)1_{n\leq V}
+\sum_{\substack{d\mid n\\d\leq U}}\mu(d)\log(n/d)
-\sum_{\substack{dc\mid n\\d\leq U,\ c\leq V}}\mu(d)\Lambda(c)
+\sum_{\substack{dcm=n\\d>U,\ c>V}}\mu(d)\Lambda(c).
$$
To prove it, use $\mu_{>U}*\mathbf1=\epsilon-\mu_{\leq U}*\mathbf1$ and $\mathbf1*\Lambda_{>V}=\log-\mathbf1*\Lambda_{\leq V}$. This gives
$$
\mu_{>U}*\mathbf1*\Lambda_{>V}
=\Lambda_{>V}-\mu_{\leq U}*\log+\mu_{\leq U}*\mathbf1*\Lambda_{\leq V},
$$
and rearrangement proves the formula, as in the <Vaughan identity proof>.
The <Bombieri–Vinogradov theorem> states that for every $A>0$ there is $B>0$ such that
$$
\sum_{q\leq Q}\max_{(a,q)=1}\max_{2\leq y\leq x}\left|\psi(y;q,a)-\frac y{\phi(q)}\right|\ll_A\frac{x}{\log^Ax},\qquad Q\leq\frac{x^{1/2}}{\log^Bx}.
$$
Here $\psi(y;q,a)=\sum_{\substack{n\leq y\\n\equiv a\pmod q}}\Lambda(n)$ is the <Chebyshev function in an arithmetic progression>, and $\phi$ is the <Euler totient function>. In particular, <partial summation> gives
$$
\sum_{q\leq Q}\max_{(a,q)=1}\left|\pi(x;q,a)-\frac{\operatorname{Li}(x)}{\phi(q)}\right|\ll_A\frac{x}{\log^Ax},\qquad\operatorname{Li}(x)=\int_2^x\frac{dt}{\log t},
$$
after choosing the logarithmic saving in the weighted theorem sufficiently large and absorbing the <prime power> terms. This is the offset <logarithmic integral function>.
For the proof strategy, <Orthogonality of Dirichlet characters> converts errors in <arithmetic progressions> into <character sums of Dirichlet characters> weighted by $\Lambda$. Reduction to <primitive Dirichlet characters> is followed by the <character large sieve>, which bounds their mean square by $O((N+Q^2)\sum|a_n|^2)$ with the usual $q/\phi(q)$ weights. The <Vaughan identity> separates the weighted sums into short terms, <Type I sums> and <Type II sums>. Direct inner-sum estimates handle the <Type I sums>; the <Cauchy-Schwarz inequality> and the <large sieve> control the <Type II sums>. The <Siegel–Walfisz theorem> supplies arbitrarily strong logarithmic savings for small moduli, where an average estimate alone would not suffice. A <dyadic decomposition>, suitable $U,V$ and a sufficiently large $B$ absorb the divisor and logarithmic losses. This explains why the range is essentially $Q^2\leq x$, with logarithmic room to spare.
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