On the mantissa distribution of powers of natural and prime numbers

Autor: Bruno Massé, Dominique Schneider, Shalom Eliahou
Rok vydání: 2012
Předmět:
Zdroj: Acta Mathematica Hungarica. 139:49-63
ISSN: 1588-2632
0236-5294
DOI: 10.1007/s10474-012-0244-1
Popis: Given a fixed integer exponent r≧1, the mantissa sequences of (nr)n and of \({(p_{n}^{r})}_{n}\), where pn denotes the nth prime number, are known not to admit any distribution with respect to the natural density. In this paper however, we show that, when r goes to infinity, these mantissa sequences tend to be distributed following Benford’s law in an appropriate sense, and we provide convergence speed estimates. In contrast, with respect to the log-density and the loglog-density, it is known that the mantissa sequences of (nr)n and of \({(p_{n}^{r})}_{n}\)are distributed following Benford’s law. Here again, we provide previously unavailable convergence speed estimates for these phenomena. Our main tool is the Erdős–Turan inequality.
Databáze: OpenAIRE
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