Expected value of the smallest denominator in a random interval of fixed radius

Huayang Chen; Alan Haynes

Expected value of the smallest denominator in a random interval of fixed radius

Number Theory 2022-11-16 v2 Numerical Analysis Numerical Analysis

Authors: Huayang Chen , Alan Haynes

Abstract

We compute the probability mass function of the random variable which returns the smallest denominator of a reduced fraction in a randomly chosen real interval of radius $\delta/2$ . As an application, we prove that the expected value of the smallest denominator is asymptotic, as $\delta\rightarrow 0$ , to $(16/\pi^2)\delta^{-1/2}.$

Keywords

smallest denominator reduced fraction random interval probability mass function expected value asymptotic analysis

Cite

@article{arxiv.2109.12668,
  title  = {Expected value of the smallest denominator in a random interval of fixed radius},
  author = {Huayang Chen and Alan Haynes},
  journal= {arXiv preprint arXiv:2109.12668},
  year   = {2022}
}

Comments

9 pages, streamlined proof, radius $\delta$ from previous version replaced by $\delta/2$

Expected value of the smallest denominator in a random interval of fixed radius

Abstract

Keywords

Cite

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