Tree asymptotic densities in number theory
Roberto Conti, Pierluigi Contucci, Vitalii Iudelevich
TL;DR
The paper develops a framework to classify integers by their prime factorization towers, encoding them as nonplanar rooted trees and introducing tree zeta functions that generalize the prime zeta function. The main result gives an explicit asymptotic density for integers whose tree matches a given T, expressed via a density signature (m,k) and the tail ζ_{T'}(1/m); the approach relies on a hierarchical decomposition T = T0^k ∘ T' and analytic estimates of π_T(x). Analytic properties of ζ_T(s), including convergence and an essential singularity at s = 1/m, underpin the constants in the density formula and connect to classical results like PNT and Landau, while offering a unifying perspective that encompasses Naslund’s factorization results. The work opens avenues for uniform refinements, higher-order terms, and deeper study of the analytic structure of tree zeta functions, highlighting their potential to distinguish trees beyond the (m,k) signature.
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.