### Manack, C and Miller, SJ (2015)

#### Leading digit laws on linear Lie groups

Research in Number Theory 1:22.

**ISSN/ISBN:** Not available at this time.
**DOI:** 10.1007/s40993-015-0024-4

**Abstract:** We study the leading digit laws for the matrix entries of a linear Lie group G. For non-compact G, these laws generalize the following observations: (1) the normalized Haar measure of the Lie group ℝ+
R
is dx/x and (2) the scale invariance of dx/x implies the distribution of the digits follow Benfordâ€™s law. Viewing this scale invariance as left invariance of Haar measure, we see either Benford or power law behavior in the significands from one matrix entry of various such G. When G is compact, the leading digit laws we obtain come as a consequence of digit laws for a fixed number of components of a unit sphere. The sequence of digit laws for the unit sphere exhibits periodic behavior as the dimension tends to infinity.

**Bibtex:**

```
@Article{,
author="Manack, Corey
and Miller, Steven J.",
title="Leading digit laws on linear Lie groups",
journal="Research in Number Theory",
year="2015",
volume="1",
number="1",
pages="22",
issn="2363-9555",
doi="10.1007/s40993-015-0024-4",
url="http://dx.doi.org/10.1007/s40993-015-0024-4"
}
```

**Reference Type:** Journal Article

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