Partitions into Primes

We investigate the asymptotic behavior of the partition function $p_{\Lambda} (n)$ defined by $\sum ^{\infty }_{n=0}p_{\Lambda} (n)x^{n} =\prod ^{\infty }_{m=1}(1-x^{m})^{-\Lambda (m)}$, where $\Lambda (n)$ denotes the von Mangoldt function. Improving a result of Richmond, we show that $\log p_{\Lambda} (n)=2\sqrt {\zeta (2)n}+O(\sqrt n\exp \{-c(\log n) (\log _{2} n)^{-2/3}(\log _{3} n)^{-1/3}\})$, where $c$ is a positive constant and $\log _{k}$ denotes the $k$ times iterated logarithm. We also show that the error term can be improved to $O(n^{1/4})$ if and only if the Riemann Hypothesis holds.