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

09/26/2021

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by   Huayang Chen, et al.

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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 δ. As an application, we prove that the expected value of the smallest denominator is asymptotic, as δ→ 0, to (8√(2)/π^2)δ^-1/2.