Partition-theoretic formulas for arithmetic densities

Ken Ono, Robert Schneider, Ian Wagner

If $\gcd (r,t)=1$, then a theorem of Alladi offers the Möbius sum identity $$-\sum _{\begin{array}{c}n\ge 2\\ p_{\min }(n)\equiv r(\mathrm{mod}t)\end{array}}\mu (n)n^{-1}=\frac{1}{\phi (t)}.$$ Here $p…