= Prime number asymptotic from Mertens cancellation
{title2=$M(x)=o(x)\Longrightarrow\psi(x)=x+o(x)$}
Let $\Delta(N)=\sum_{n\le N}(\tau(n)-\log n-2\gamma)$. The <Möbius divisor-sum identity> and the <Von Mangoldt divisor identity> give $\mu*\tau=\mathbf1$ and $\mu*\log=\Lambda$, so
$$
\psi(N)=N-2\gamma-\sum_{d\le N}\mu(d)\Delta(\lfloor N/d\rfloor).
$$
Here $*$ is <Dirichlet convolution>. The <divisor summatory asymptotic> bounds the terms with $d\le N/K$ in absolute value by $O(N/\sqrt K)$. For $d>N/K$, group by $j=\lfloor N/d\rfloor<K$: the remaining sum is $\sum_{j<K}\Delta(j)(M(N/j)-M(N/(j+1)))=o_K(N)$. First let $N$ tend to infinity, then $K$ tend to infinity. This proves the <Second Chebyshev function> asymptotic from cancellation of the <Mertens function>.
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