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⊆N, the series −∑μ𝒫(λ)q|λ| over partitions into distinct parts with smallest part in S tends to the density of S as q→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/log x]; natural density does not suffice. Multiplied by log(1/(1−q)), the series tends to δ if and only if S has (cδ+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 eΨβ/t with Ψβ>0. Along q=e−t(1+iβ), the formula for residue classes modulo m≥3 holds exactly when |β|<βm, an explicit Clausen-function threshold with βm→0, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.
Related
- Probability-theoretic interpretation of degeneracy locus formulas
- Distributional Fractional Taylor Series and Interpretation of the Fractal and Number-Theoretic Explicit Formulas
- L-tauberian operators and subspaces with equi-integrable unit ball
- Tauberian rates and the Baire property of ideals
- A Classification of Fourier Summation Formulas and Crystalline Measures
BibTeX
@misc{marashdeh2026partitiontheoretic,
title = {{Partition-theoretic density formulas: radial Tauberian theory and non-radial instability}},
author = {Mohammad F. Marashdeh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10570},
eprint = {2610.10570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10570},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Partition-theoretic density formulas: radial Tauberian theory and non-radial instability
AU - Mohammad F. Marashdeh
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.10570
UR - https://arxiv.org/abs/2610.10570
AB - Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets S⊆N, the series −∑μ_𝒫(λ)q^|λ| over partitions into distinct parts with smallest part in S tends to the density of S as q→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/log x]; natural density does not suffice. Multiplied by log(1/(1−q)), the series tends to δ if and only if S has (cδ+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^Ψ_β/t with Ψ_β>0. Along q=e^−t(1+iβ), the formula for residue classes modulo m≥3 holds exactly when |β|<β_m, an explicit Clausen-function threshold with β_m→0, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.
ER -