{"title":"The automaticity of the set of primes","authors":"Thomas Dubbe","doi":"arxiv-2409.04314","DOIUrl":null,"url":null,"abstract":"The automaticity $A(x)$ of a set $\\mathcal{X}$ is the size of the smallest\nautomaton that recognizes $\\mathcal{X}$ on all words of length $\\leq x$. We\nshow that the automaticity of the set of primes is at least\n$x\\exp\\left(-c(\\log\\log x)^2\\log\\log\\log x\\right)$, which is fairly close to\nthe maximal automaticity.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"9 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.04314","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The automaticity $A(x)$ of a set $\mathcal{X}$ is the size of the smallest
automaton that recognizes $\mathcal{X}$ on all words of length $\leq x$. We
show that the automaticity of the set of primes is at least
$x\exp\left(-c(\log\log x)^2\log\log\log x\right)$, which is fairly close to
the maximal automaticity.
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