Optimal Selberg weights have modulus at most one
= Optimal Selberg weights have modulus at most one
Let $k(d)=\prod_{p\mid d}g(p)/(1-g(p))$ on permitted <squarefree integers>. The <Selberg sieve weights> have
$$
|\rho_d|=\frac1{J\prod_{p\mid d}(1-g(p))}\sum_{\substack{v\leq L/d\\(v,d)=1}}k(v)\leq1.
$$
The numerator is the total weight of the distinct integers $ev\leq L$ with $e\mid d$ and $(v,d)=1$: <multiplicativity> gives $\sum_{e\mid d}k(e)=\prod_{p\mid d}(1-g(p))^{-1}$. They are a subset of the terms in $J$.