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

Abstract

Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets $S\subseteq\N$, the series $-\sum\mu_{\mathcal P}(\lambda)q^{|\lambda|}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\to1$. 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 $(x,x+cx/\log x]$; natural density does not suffice. Multiplied by $\log(1/(1-q))$, the series tends to $\delta$ if and only if $S$ has $(c\delta+o(1))x/(\log x)^2$ 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 $e^{\Psi_\beta/t}$ with $\Psi_\beta>0$. Along $q=e^{-t(1+i\beta)}$, the formula for residue classes modulo $m\ge3$ holds exactly when $|\beta|<\beta_m$, an explicit Clausen-function threshold with $\beta_m\to0$, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.