Completely Additive Height Functions: Profile Laws, Matula Bounds, and Inverse Growth
Hartosh Singh Bal
TL;DR
This work introduces height functions $H: \mathbb{N}\to\mathbb{N}_0$ that are completely additive with finite prime fibres, and shows that the resulting prime-height profile $(\pi_k)$ completely determines the height multiplicities $N_n$ via the Euler product $\sum_{n\ge0}N_n q^n=\prod_{k\ge1}(1-q^k)^{-\,\pi_k}$. It builds a broad framework of iteratively defined and squarefree height variants, including Matula and Shapiro-type heights, and establishes a correspondence between these profiles and multipartition structures, linking number theory to partition theory. In the polynomial regime, the authors prove an inverse-growth principle: if the cumulative profile satisfies $\Pi(x)\sim (C/\alpha)x^{\alpha}$, then $\log N_n$ grows like a stretched exponential with exponent $\alpha/(\alpha+1)$, with explicit constants, and conversely under Tauberian hypotheses. They further introduce a canonical sequential realisation to study average orders of heights in this regime, compute explicit benchmark cases for ordinary and plane partitions, and supply substantial computational evidence—especially for the Shapiro-type height—that primes within a fixed height window exhibit lognormal fluctuations in $\log p$. The results illuminate a robust, profile-driven landscape for height functions, with potential extensions to other multiplicative systems and regimes beyond polynomial growth; they also suggest rich horizontal structure within height layers and several inviting directions for future work.
Abstract
The height $H(n)$ of $n$ is the least integer $i$ such that the $i$-th iterate of Euler's totient function $\varphi^{(i)}(n)$ equals $1$. H. N. Shapiro showed that this $H$ is almost completely additive. Building on the fact that this function can be modified to yield a completely additive function, we establish a general correspondence: to every multi-partition structure there corresponds a completely additive function. In this paper, a \emph{height function} is a completely additive map $H:\mathbb{N}\to\mathbb{N}_0$ with $H(1)=0$ whose prime fibres $\{p:\,H(p)=k\}$ are finite for every $k\ge1$. Writing \[ π_k=\#\{p:\,H(p)=k\},\qquad N_k=\#\{n:\,H(n)=k\}, \] complete additivity forces the identity \[ \sum_{k\ge0}N_k q^k \;=\; \prod_{j\ge1}(1-q^j)^{-π_j}. \] Thus, the prime--height profile $(π_k)$ canonically determines the height multiplicities $(N_k)$, linking to the asymptotic theory of weighted partitions. We introduce a broad class of iteratively defined heights on primes, encompassing Matula-type heights (encoding rooted trees) and Shapiro-type totient heights, and show they extend to genuine height functions. In the Matula case this yields a purely number-theoretic proof of the classical extremal bounds for minimal and maximal Matula numbers, answering a question of Gutman and Ivić without recourse to graph theory. Using Meinardus' theorem we prove an \emph{inverse-growth} principle in the polynomial regime: if $Π(x)=\sum_{j\le x}π_j \sim (C/α)x^α$, then $\log N_k$ satisfies a stretched-exponential law with an explicit constant, and conversely under a standard Tauberian hypothesis. We further derive average-order consequences in this regime for a canonical sequential realization of a given profile. Finally, we briefly discuss behavior beyond the polynomial setting, with computations in the Shapiro case suggesting substantially richer phenomena.