Multiplication Tables for Integers with Restricted Prime Factors
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908902616989696 |
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| author | Schlitt, Jeremy |
| author_facet | Schlitt, Jeremy |
| contents | Let $Q$ be a set of primes with relative density $δ$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $δ\in (0,1]$. This generalizes the case $δ= 1$ from the 2008 work of Ford. We also show that there is a phase transition at the critical point $δ= 1/\log 4$, for which we explicitly determine the behaviour. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19212 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiplication Tables for Integers with Restricted Prime Factors Schlitt, Jeremy Number Theory Probability 11N25 (Primary) 11N36, 60G50 (Secondary) Let $Q$ be a set of primes with relative density $δ$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $δ\in (0,1]$. This generalizes the case $δ= 1$ from the 2008 work of Ford. We also show that there is a phase transition at the critical point $δ= 1/\log 4$, for which we explicitly determine the behaviour. |
| title | Multiplication Tables for Integers with Restricted Prime Factors |
| topic | Number Theory Probability 11N25 (Primary) 11N36, 60G50 (Secondary) |
| url | https://arxiv.org/abs/2603.19212 |