Divisor summatory asymptotic (${\sum }_{n\le x}\tau (n)=x\log x+(2\gamma -1)x+O(\sqrt{x})$)

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 $N$.