Partition-theoretic density formulas: radial Tauberian theory and non-radial instability

Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi’s duality formula: for many sets $S\subseteq \mathbb{N}$, the series $-\sum {\mu }_{\mathcal{P}}(\lambda )q^{\mathrm{\mid }\lambda \mathrm{\mid }}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\to 1$. We determine when this happens. Radially, the summation kernel is a Gumbel density, and Wiener’s theorem shows that for bounded $f$ the associated series tends to $L$ if and only if $f$ has mean $L$ on every interval $(x,x+cx\mathrm{/}\log x]$; natural density does not suffice. Multiplied by $\log (1\mathrm{/}(1-q))$, the series tends to $\delta$ if and only if $S$ has $(c\delta +o(1))x\mathrm{/}(\log x{)}^2$ elements in each such interval; this gives a partition analogue of Alladi’s theorem for Chebotarev sets of primes. Off the real axis the largest weights grow like $e^{{\mathrm{\Psi }}_{\beta }\mathrm{/}t}$ with ${\mathrm{\Psi }}_{\beta }>0$. Along $q=e^{-t(1+i\beta )}$, the formula for residue classes modulo $m\ge 3$ holds exactly when $\mathrm{\mid }\beta \mathrm{\mid }<{\beta }_m$, an explicit Clausen-function threshold with ${\beta }_m\to 0$, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.