# On the asymptotic distinct prime partitions of integers

## TL;DR

This paper derives an improved asymptotic formula for the number of ways to write an integer as a sum of distinct primes, including higher-order corrections, which significantly enhances approximation accuracy.

## Contribution

It introduces a refined asymptotic expression for $Q(n)$ using saddle-point approximation and higher-order corrections, surpassing previous simple exponential estimates.

## Key findings

- The new asymptotic form closely matches the exact $Q(n)$ for large $n$.
- Higher-order corrections improve the approximation accuracy.
- The method involves Laplace inversion of the fermionic partition function.

## 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.