A remark on weighted average multiplicities in prime factorisation

TL;DR

The paper introduces a meromorphic family of weighted average multiplicities $\mathfrak{W}(n,s)$ that generalizes the ABC-geometry-like weighting of prime factors by allowing complex exponents $s$. It proves divergence of $\mathfrak{W}(n,s)$ for $\Re(s)<1$ over ABC triples in both the integer and polynomial settings, and shows boundedness for $\Re(s)$ beyond a critical boundary $a_{\text{crit}}$, with $a_{\text{crit}}$ determined by an equality of logarithmic terms involving the largest prime factor. The work combines explicit constructions (e.g., $(1,2^n-1,2^n)$ and $(1,x^p-1,x^p)$) with counting arguments in finite fields, and employs analytic tools like Jensen's inequality and triangle bounds to study pole structure and asymptotics as $\Re(s)\to+\infty$. Computational experiments on large $abc$ datasets reveal intricate pole patterns near $\Re(s)=1$ and support the identification of $a_{\text{crit}}$ as a natural boundary for analyticity, connecting the distribution of the largest prime factor to the ABC conjecture through $\mathfrak{W}(n,s)$.

Abstract

We study a generalisation of the quality of an ABC triple that we call the weighted average multiplicity (WAM), in which the logarithmic heights of prime factors are raised to a complex exponent s. The WAM is connected to the standard ABC conjecture at s=1. We show that for real part of s less than 1, WAM is unbounded over ABC triples both for integers and polynomials. For real part greater than 1, we characterise a boundary beyond which WAM is holomorphic and bounded. In this region, we show that WAM is related to the multiplicity of the largest prime factor of the triple, a quantity that we connect with the original ABC conjecture and whose distribution we explore computationally.

Figures (5)

• Figure 1: Values of $\log_{10} \max \lvert\mathop{\mathrm{\mathfrak{W}}}\nolimits(abc,s)\rvert$ as a function of $s$ in the complex plane, where $\max$ is taken over ABC triples with $c\sim10^{18}$. Bright colors are poles of $\mathop{\mathrm{\mathfrak{W}}}\nolimits$ for individual $n$, which are then accumulated with $\max$.
• Figure 2: Values of $\mathop{\mathrm{\mathfrak{W}}}\nolimits$ on the line $\Re(s) = 1$. Chaotic behaviour occurs due to poles of $\mathop{\mathrm{\mathfrak{W}}}\nolimits$.
• Figure 3: Number of triples with a specific multiplicity of the largest prime factor $e_m$ in the dataset.
• Figure 4: Numerically computed values of $a_\text{crit}$.
• Figure 5: Numerically found pole locations for six ABC triples. Each sub-figure shows the scatter plot $(a,b)$, critical vertical line at $a_{\rm crit}$ and the corresponding $abc$-triple.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Proposition 1
• Lemma 2: Equidistribution
• Lemma 3
• proof
• Proposition 2
• proof