A remark on weighted average multiplicities in prime factorisation
Number Theory
2025-10-09 v1
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.
Keywords
Cite
@article{arxiv.2510.06993,
title = {A remark on weighted average multiplicities in prime factorisation},
author = {Viktor Mirjanić and Daattavya Aggarwal and Challenger Mishra},
journal= {arXiv preprint arXiv:2510.06993},
year = {2025}
}
Comments
19 pages, 5 figures