Explicit Bounds for the Approximation Error in Benford's Law

@article{Duembgen2007ExplicitBF,
  title={Explicit Bounds for the Approximation Error in Benford's Law},
  author={Lutz Duembgen and Christoph Leuenberger},
  journal={Electronic Communications in Probability},
  year={2007},
  volume={13},
  pages={99-112}
}
Benford's law states that for many random variables $X > 0$ its leading digit $D = D(X)$ satisfies approximately the equation $\mathbb{P}(D = d) = \log_{10}(1 + 1/d)$ for $d = 1,2,\ldots,9$. This phenomenon follows from another, maybe more intuitive fact, applied to $Y := \log_{10}X$: For many real random variables $Y$, the remainder $U := Y - \lfloor Y\rfloor$ is approximately uniformly distributed on $[0,1)$. The present paper provides new explicit bounds for the latter approximation in terms… 
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