= Selberg sieve denominator asymptotic
{c}
{title2=$G(z)=\sum_{n\le z}\mu(n)^2/\varphi(n)=\log z+O(1)$}
Put $g(n)=\mu(n)^2/\varphi(n)$. As a <Dirichlet convolution>, $g=h*(n\mapsto1/n)$, where $h$ is a <multiplicative arithmetic function> with $h(p)=1/(p(p-1))$, $h(p^2)=-1/(p(p-1))$, and $h(p^j)=0$ for $j\ge3$. The local sums are one, so $\sum_nh(n)=1$. The series $\sum_n|h(n)|\log(2n)$ converges, since the prime-local errors are $O((\log p)/p^2)$. Consequently $G(z)=\sum_{d\le z}h(d)H_{\lfloor z/d\rfloor}=\log z+O(1)$ by the <harmonic number> estimate $H_m=\log m+O(1)$. This identifies the leading constant in the one-dimensional <Selberg upper-bound sieve>.
Back to article page
