Acta Mathematica Hungarica

, Volume 139, Issue 1, pp 49–63

On the mantissa distribution of powers of natural and prime numbers

  • Shalom Eliahou
  • Bruno Massé
  • Dominique Schneider
Article

DOI: 10.1007/s10474-012-0244-1

Cite this article as:
Eliahou, S., Massé, B. & Schneider, D. Acta Math Hung (2013) 139: 49. doi:10.1007/s10474-012-0244-1

Abstract

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–Turán inequality.

Key words and phrases

Benford’s lawmantissaprime number

Mathematics Subject Classification

60B1011B0511K99

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  • Shalom Eliahou
    • 1
    • 2
    • 3
  • Bruno Massé
    • 1
    • 2
    • 3
  • Dominique Schneider
    • 1
    • 2
    • 3
  1. 1.ULCOLMPA J. LiouvilleCalaisFrance
  2. 2.Univ. Lille Nord de FranceLilleFrance
  3. 3.CNRSParisFrance