An improvement on the parity of Schur's partition function

TL;DR

This paper refines the bounds on the distribution of odd values of Schur's partition function, providing a tighter estimate on how often these values are odd within a range.

Contribution

It improves previous bounds on the parity distribution of Schur's partition function, offering a more precise asymptotic estimate.

Findings

Enhanced bounds on the count of odd values of A(2n+1)

Tighter asymptotic estimates involving logarithmic factors

Advancement over previous results by Chen

Abstract

We improve S.-C. Chen's result on the parity of Schur's partition function. Let $A(n)$ be the number of Schur's partitions of $n$, i.e., the number of partitions of $n$ into distinct parts congruent to $1, 2 \mod{3}$. S.-C. Chen \cite{MR3959837} shows $\small \frac{x}{(\log{x})^{\frac{47}{48}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}$. In this paper, we improve Chen's result to $\frac{x}{(\log{x})^{\frac{11}{12}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}.$