An Elementary Analysis of the Prime Partition Function

Abstract

Let $pp (n)$ denote the number of ways to write $n$ as a sum of primes. In this paper, we show that $$ \log pp (n) \sim 2π\sqrt{\frac{n}{3\log n}}\;. $$ While sharper estimates are already known, they rely on highly involved and lengthy proofs. In sharp contrast, our approach uses a short, elementary recipe that easily adapts to yield similar asymptotic estimates for several related, extensively studied problems.