Some remarks on the first digit problem.

*(English)*Zbl 0912.60001The problem of the first significant digit can be described as follows: in a very large sample of positive real numbers, randomly chosen, if the frequency of the first non-zero digit is tested, it appears that the occurrence of the 9 digits is not the same for all. The experiments so far performed agree with the so-called “Benford law”: if \(f(j)\) denotes the frequency of the digit \(j\), then \(f(j)= \text{Log} (j+1) -\text{Log} (j)\), with \(\text{Log} =\log_{10}\). The typical way to approach the first-digit problem is to denote by \(Y\) the “random variable” resulting from the experiments, i.e. \(Y\) is any positive real number, completely randomly chosen, and to describe the distribution of \(Y\), in order to deduce under suitable conditions that the first significant digit of \(Y\) obeys the Benford law. The main purpose of the present paper is to describe the “distribution” of \(Y\), first introducing some models and then deducing the Benford-compatibility from some kinds of invariance. Section 1 of the paper introduces the classes of functions that satisfy suitable limit conditions and relates these properties to weak*-convergence of an appropriate net of measurable functions. Section 2 examines the convergence to Benford-compatible distributions for suitable sequences \((Y_n)\) associated to \(Y\). Section 3 defines Benford-compatibility and suggests a model for \(Y\) as a limit of suitable sequences \((Y_n)\). Finally, two different types of invariance are defined, power-invariance and \(r\)-scale-invariance, and it is proved that power-invariance is equivalent to Benford-compatibility, while \(r\)-scale-invariance is equivalent to a weaker form of Benford-compatibility.

Reviewer: N.Curteanu (Iaşi)