A remark on weighted average multiplicities in prime factorisation
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914081438433280 |
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| author | Mirjanić, Viktor Aggarwal, Daattavya Mishra, Challenger |
| author_facet | Mirjanić, Viktor Aggarwal, Daattavya Mishra, Challenger |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06993 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A remark on weighted average multiplicities in prime factorisation Mirjanić, Viktor Aggarwal, Daattavya Mishra, Challenger Number Theory 11D75 (Primary) 11N37, 30D30 (Secondary) 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. |
| title | A remark on weighted average multiplicities in prime factorisation |
| topic | Number Theory 11D75 (Primary) 11N37, 30D30 (Secondary) |
| url | https://arxiv.org/abs/2510.06993 |