On Benford's law to variable base (Q1387689)

Given two real numbers \(y\neq 0\) and \(b>1\), we can write \(| y|= mb^k\), where \(1\leq m<b\) and \(k\) is an integer. Here \(m=m(y,b)\) is called the mantissa of \(y\), while \(k=k (y,b)\) is called the exponent of \(y\) to the base \(b\). If \(y\) is a random variable, then the distribution function \(M(b,x): =P[m(y,b) <x]\) is called the mantissa distribution of \(y\) to the base \(b\). In a number of cases, \(m\) is logarithmically distributed: \(M(b,x)= \log_bx\) for \(1<x <b\). This phenomenon is called Benford's law to the base \(b\). From the main result of the present paper it follows that the approximation by Benford's law becomes worse as \(b\to\infty\), but it gets better as \(b\to 1+0\).