### Eliahou, S, Massé, B and Schneider, D (2013)

#### On the mantissa distribution of powers of natural and prime numbers

Acta Mathematica Hungarica, 139(1), pp. 49-63.

**ISSN/ISBN:** 0236-5294
**DOI:** 10.1007/s10474-012-0244-1

**Abstract:** Given a fixed integer exponent r≧1, the mantissa sequences of (n r ) n and of , 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 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.

**Bibtex:**

```
@article {,
AUTHOR = {Eliahou, Shalom and Massé, Bruno and Schneider, Dominique},
TITLE = {On the mantissa distribution of powers of natural and prime numbers},
JOURNAL = {Acta Mathematica Hungarica},
YEAR = {2012},
MONTH = {June},
PAGES = {1--15},
ISSN = {0236-5294},
DOI = {10.1007/s10474-012-0244-1},
URL = {http://link.springer.com/article/10.1007/s10474-012-0244-1},
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Number Theory