Tree asymptotic densities in number theory

TL;DR

This paper investigates the asymptotic distribution of integers based on their prime factorization trees, deriving explicit density formulas and introducing a generalized tree zeta function that encompasses classical number theory results.

Contribution

It introduces a novel framework linking prime factorization trees to density formulas and generalizes the prime zeta function to a new tree zeta function.

Findings

Derived explicit density formulas for various trees

Connected classical results like the prime number theorem as special cases

Introduced the tree zeta function generalizing the prime zeta function

Abstract

We study the asymptotic distribution of integers sharing the same rooted-tree structure that encodes their complete prime factorization tower. For each tree we derive an explicit density formula depending only on a pair $(m,k)$, the density signature of the tree, up to a suitable multiplicative scalar factor and introduce the corresponding tree zeta function, which generalizes the prime zeta function. Classical results such as the prime number theorem and later work by Landau appear as special cases.