On the asymptotic distinct prime partitions of integers

Abstract

We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to \infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple leading-order exponential form given so far in the literature.

arXiv comment10 pages, 3 figures; v2: small editorial changes, correction of misprints v3: revised version, added new material by V. Kotesovec that confirms our conclusions