On the mantissa distribution of powers of natural and prime numbers

Products of random variables and the first digit phenomenon

“…This led him to make a scale-invariance hypothesis (more or less satisfied in real life) from which he derived that µ 10 can be seen as the (ideal) distribution of digits or mantissa of many real-life numbers. Of course, this ideal distribution is never achieved in practice.Several mathematicians have been involved in this subject and have provided sequences of positive numbers whose mantissae are (or approach to be) distributed following µ b in the sense of the natural density [1,6,8,11,23] (see Definition 2.1), random variables whose mantissa law is or approaches µ b [3,10,15,18,21], sequences of random variables whose mantissae laws converge to µ b or whose mantissae are almost surely distributed following µ b [22,24,29,28]. Among the many applications of the First Digit Phenomenon, we can quote: fraud detection [26], computer design [14,19] (data storage and roundoff errors), image processing [30] and data analysis in natural sciences [25,27].…”

On the discrepancy of powers of random variables

“…In a different context, such a convergence was already observed in Theorem 1 in [7], in which it is stated that two (deterministic) sequences at a large power tend to be Benford.…”

Fast growing sequences of numbers and the first digit phenomenon

“…Proof. Lemma 2 in [5] says that with θ = ahl/ log b. So, by (3.1), the sequences (log b (U n+h /U n )) n (h ≥ 1) satisfy the hypotheses of Lemma 2.10 because θ → ∞ when h → ∞.…”

“…Their mantissa is distributed following Benford's law though, but in a weaker sense than in (1.1) [3,5,10]. Namely, if (v n ) n is one of these sequences, then the sequences ( 1 The sequence (log n) n is neither natural-Benford nor log-Benford [7].…”