Singular series ($\mathfrak{S}(n)$)

A singular series assembles the local congruence densities in an additive counting asymptotic. For three prime summands it is ${\sum }_{q\ge 1}\mu (q)c_q(n)/\phi (q{)}^3$, 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.