H.F.D. ($H$-function distribution) and the Benford law. I

This paper notes a connection among a wide class of the so-called $HF$-random variables, approximately uniform distributions, and Benford's law. This connection is considered in detail with the help of examples of random variables having gamma-distribution. Let $Y$ be a random variable having gamma-distribution with parameter $\alpha$. It is proved that the distribution of a fractional part of the logarithm of $Y$ with respect to any base larger than 1 converges to the uniform distribution on the interval $[0,1]$ for $\alpha\to0$. This implies that the probability distribution of the first significant digit of $Y$ for small $\alpha$ can be approximately described by Benford's law. The order of the approximation is illustrated by tables.