On the mantissa distribution of powers of natural and prime numbers

Abstract

Given a fixed integer exponent r≧1, the mantissa sequences of (n r) n and of \({(p_{n}^{r})}_{n}\), where p n 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 (n r) 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.

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Correspondence to Bruno Massé.

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Eliahou, S., Massé, B. & Schneider, D. On the mantissa distribution of powers of natural and prime numbers. Acta Math Hung 139, 49–63 (2013). https://doi.org/10.1007/s10474-012-0244-1

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Key words and phrases

  • Benford’s law
  • mantissa
  • prime number

Mathematics Subject Classification

  • 60B10
  • 11B05
  • 11K99