= Divisor summatory asymptotic
{title2=$\sum_{n\le x}\tau(n)=x\log x+(2\gamma-1)x+O(\sqrt x)$}
The <Dirichlet hyperbola method> gives $D(x)=2\sum_{a\le r}\lfloor x/a\rfloor-r^2$, where $r=\lfloor\sqrt x\rfloor$. The <harmonic number> estimate $H_r=\log r+\gamma+O(1/r)$, with $\gamma$ the <Euler's constant>, yields $D(x)=2x(\log r+\gamma)-r^2+O(x/r+r)$. Substitution of $r=\sqrt x+O(1)$ gives the displayed <asymptotic expansion>. Comparison of $\sum_{n\le x}\log n$ with its integral consequently gives $\sum_{n\le N}(\tau(n)-\log n-2\gamma)=O(\sqrt N)$ for integer $N$.
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