On the asymptotic prime partitions of integers

TL;DR

This paper derives an analytical asymptotic formula for the number of prime partitions of integers using quantum statistical mechanics techniques, improving previous estimates by including higher-order corrections and comparing with numerical data.

Contribution

It introduces a novel analytical approach to prime partitions using quantum statistical mechanics, providing higher-order asymptotic corrections and validating them against large-scale numerical data.

Findings

Asymptotic growth of prime partitions is exponential in sqrt(n/ln(n)).

Next-to-leading order correction to the asymptotic formula is computed.

Higher-order correction term is derived, not previously available in literature.

Abstract

In this paper, we discuss P(n), the number of ways in which a given integer n may be written as a sum of primes. In particular, an asymptotic form P_as(n) valid for n towards infinity is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This directly gives the asymptotic number of prime partitions P_as(n). The leading term in the asymptotic expression grows exponentially as sqrt[n/ln(n)] and agrees with previous estimates. We calculate the next-to-leading order term in the exponent, porportional to ln[ln(n)]/ln(n), and show that an earlier result in the literature for its coefficient is incorrect. Furthermore, we also calculate the next higher order correction, proportional to 1/ln(n) and given in Eq.(43), which so far has not been available in the literature. Finally, we compare our analytical results with the exact numerical values of P(n) up to n \sim 8 10^6. For the highest values, the remaining error between the exact P(n) and our P_as(n) is only about half of that obtained with the leading-order (LO) approximation. But we also show that, unlike for other types of partitions, the asymptotic limit for the prime partitions is still quite far from being reached even for n \sim 10^7.