On the number of parts in congruence classes for partitions into distinct parts

For integers $$0 < r \le t$$ , let the function $$D_{r,t}(n)$$ denote number of parts among all partitions n into distinct that are congruent to modulo t. We prove asymptotic formula $$\begin{aligned} D_{r,t}(n) \sim \dfrac{3^{\frac{1}{4}} e^{\pi \sqrt{\frac{n}{3}}}}{2\pi t n^{\frac{1}{4}}} \left( \log (2) + \dfrac{\sqrt{3} (2)}{8\pi } - \dfrac{\pi }{4\sqrt{3}} \dfrac{t}{2} \right) n^{- \frac{1}{2}} \end{aligned}$$ as $$n \rightarrow \infty $$ . A corollary this result is for $$0< s inequality $$D_{r,t}(n) \ge D_{s,t}(n)$$ holds sufficiently large n. make effective, showing $$2 10$$ > 8$$