= Selberg sieve diagonalization
{c}
Put $L=\sqrt D$, $h(t)=\prod_{p\mid t}(g(p)^{-1}-1)$ and $y_t=\sum_{\substack{m\leq L\\t\mid m}}g(m)\rho_m$, where $g$ is a <multiplicative arithmetic function> supported on <squarefree integers>. For <squarefree integers> $m,n$,
$$
g([m,n])=g(m)g(n)\sum_{t\mid(m,n)}h(t).
$$
Thus the quadratic form is $\Sigma=\sum_{t\leq L}h(t)y_t^2$. <Möbius inversion> gives $g(d)\rho_d=\sum_{d\mid t\leq L}\mu(t/d)y_t$. On the permitted <squarefree integers>, the constraint $\rho_1=1$ becomes $\sum_t\mu(t)y_t=1$. The <Cauchy-Schwarz inequality> yields $\Sigma\geq1/J$, where $J=\sum_t1/h(t)$ over those integers; equality holds at $y_t=\mu(t)/(Jh(t))$.
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