= Elementary lower bound for the twin-prime sieve denominator
{title2=$\sum_{\substack{d\le z\\d\text{ squarefree}}}\prod_{p\mid d}\frac{\rho(p)}{p-\rho(p)}\gg(\log z)^2$}
For the <polynomial> $F(n)=n(n+2)$, take $\rho(2)=1$ and $\rho(p)=2$ at odd <primes>. On odd <squarefree integers>, the summand is at least $2^{\omega(d)}/d$. Put $y=\sqrt z$ and $L(y)=\sum_{a\le y,\ a\text{ odd}}1/a=\tfrac12\log y+O(1)$. The weight of odd pairs $a,b\le y$ is $L(y)^2$. The pairs for which $a$ is not <squarefree>, $b$ is not <squarefree>, or $(a,b)>1$ each have weight at most $L(y)^2\sum_{p\text{ odd}}p^{-2}$. Indeed, the weighted count of multiples of an odd $p^k$ is $p^{-k}L(y/p^k)\le p^{-k}L(y)$. Moreover,
$$
\sum_{p\text{ odd}}p^{-2}\le\sum_{m\ge1}(2m+1)^{-2}\le\frac19+\int_1^\infty\frac{dt}{(2t+1)^2}=\frac5{18}.
$$
Hence the odd <coprime> <squarefree> pairs have weight at least $L(y)^2/6$. Their products $d=ab\le z$ are <squarefree>, and $2^{\omega(d)}$ counts all ordered <coprime> factorizations of $d$. This proves the displayed lower bound needed for the <Selberg upper-bound sieve>.
Back to article page
