Higher Order Turan Inequalities for the Distinct Partition Function

TL;DR

This paper proves that the partition function into distinct parts satisfies higher order Turán inequalities for sufficiently large n, confirming conjectures and providing explicit error bounds based on asymptotic formulas.

Contribution

It establishes the validity of higher order Turán inequalities for the distinct partition function for large n, with explicit error estimates.

Findings

q(n) is log-concave for n ≥ 33

q(n) satisfies higher order Turán inequalities for n ≥ 121

Explicit error bounds are derived using Chern's asymptotic formulas

Abstract

We prove that the number $q(n)$ of partitions into distinct parts is log-concave for $n \geq 33$ and satisfies the higher order Tur\'an inequalities for $n\geq 121$ conjectured by Craig and Pun. In doing so, we establish explicit error terms for $q(n)$ and for $q(n-1)q(n+1)/q(n)^2$ based on Chern's asymptotic formulas for $\eta$-quotients.