Möbius harmonic logarithmic moments ($A_j(x)$)

Put $A_j(x)={\sum }_{d\le x}\mu (d){\log }^j(x/d)/d$. The elementary estimates areThe identities ${\sum }_d\mu (d)\lfloor x/d\rfloor =1$ and ${\sum }_d\mu (d)H_{\lfloor x/d\rfloor }/d=1$ give the first two bounds. Convolve the harmonic divisor sum with $\mu (d)/d$: because $\mu \ast \tau =1$, its left side is $H_{\lfloor x\rfloor }$. Thus $\frac{1}{2}{A}_2+2\gamma A_1+c{A}_0=\log x+O(1)$. The summed error is $O(x^{-1/2}{\sum }_{d\le x}{d}^{-1/2}\log (2x/d))=O(1)$.

This elementary smoothing argument is used in Selberg's original proof and Ramaré's sieve lectures, Lemma 4.2.