 Solution

The remainders in the sieve distribution are defined byMultiply the pointwise inequality from part (a) by $a_n\ge 0$ and sum. Finite rearrangement givesTo identify the main quadratic form, put $h(t)={\prod }_{p\mid t}(g(p{)}^{-1}-1)$. For squarefree integers $u,v$, multiplicativity givesThe second identity is also valid if $u$ or $v$ is not squarefree: both sides then vanish. Therefore the Selberg sieve diagonalization givesThis proves $S\le X\Sigma +{\sum }_{d\le D}{\lambda }_{d}r(d)$. Notice that the hypotheses do not force ${\rho }_m$ to vanish on non-squarefree integers; those coefficients simply make no contribution to this main quadratic form because $g(m)=0$.