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arXiv · 2510.06993

A remark on weighted average multiplicities in prime factorisation

Viktor Mirjanić·Daattavya Aggarwal·Challenger Mishra
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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.

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Viktor Mirjanić, Daattavya Aggarwal, Challenger Mishra. 2025-10-08. A remark on weighted average multiplicities in prime factorisation. https://arxiv.org/abs/2510.06993

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