An improvement on the parity of Schur's partition function

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 $1,2\mathrm{m}\mathrm{o}\mathrm{d}3$. S.-C. Chen \cite{MR3959837} shows $\frac{x}{(\log x{)}^{\frac{47}{48}}}\ll \mathrm{♯}\{0\le n\le x:A(2n+1)\text{is odd}\}\ll \frac{x}{(\log x{)}^{\frac{1}{2}}}$. In this paper, we improve Chen’s result to $\frac{x}{(\log x{)}^{\frac{11}{12}}}\ll \mathrm{♯}\{0\le n\le x:A(2n+1)\text{is odd}\}\ll \frac{x}{(\log x{)}^{\frac{1}{2}}}\mathrm{.}$