## Abstract

For a set of positive integers A, let p_{A}(n) denote the number of ways to write n as a sum of integers from A, and let p(n) denote the usual partition function. In the early 40s, Erdős extended the classical Hardy–Ramanujan formula for p(n) by showing that A has density α if and only if logp_{A}(n)∼logp(αn). Nathanson asked if Erdős's theorem holds also with respect to A's lower density, namely, whether A has lower-density α if and only if logp_{A}(n)/logp(αn) has lower limit 1. We answer this question negatively by constructing, for every α>0, a set of integers A of lower density α, satisfying [Formula presented] We further show that the above bound is best possible (up to the o_{α}(1) term), thus determining the exact extremal relation between the lower density of a set of integers and the lower limit of its partition function. We also prove an analogous theorem with respect to the upper density of a set of integers, answering another question of Nathanson.

Original language | English |
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Article number | 108554 |

Journal | Advances in Mathematics |

Volume | 407 |

DOIs | |

State | Published - 8 Oct 2022 |

## Keywords

- Elementary proofs
- Number theory
- Partition function

## All Science Journal Classification (ASJC) codes

- General Mathematics