Singular series
= Singular series
{title2=$\mathfrak S(n)$}
A singular series assembles the local congruence densities in an additive counting asymptotic. For three prime summands it is $\sum_{q\ge1}\mu(q)c_q(n)/\varphi(q)^3$, with <Ramanujan sum> $c_q(n)$. Its Euler product vanishes for even $n$ and has a uniform positive lower bound for odd $n$. Different additive problems have different local factors, so this is a specific instance rather than a universal formula.