# zbMATH — the first resource for mathematics

Benford’s law for the $$3x+1$$ function. (English) Zbl 1117.11018
The $$3x+1$$ problem (sometimes called Collatz-problem) concerns the behaviour under iteration of the function $$T:\mathbb{N}\to\mathbb{N}$$ defined by $$T(n)= \frac n2$$ if $$n$$ is even and $$T(n)=\frac{3n+1}{2}$$ if $$n$$ is odd. The $$3x+1$$ conjecture asserts that when started from any $$n \in\mathbb{N}$$ some $$k$$ exists with iterate $$T^{(k)}(n)=1$$ [see also J. C. Lagarias, Am. Math. Mon. 92, 3–23 (1985; Zbl 0566.10007)].
Benford’s law [F. Benford, Proc. Am. Philos. Soc. 78, 551–572 (1938; Zbl 0018.26502)] for an infinite sequence $$\{x_k \mid k\geq 1\}$$ of positive quantities $$x_k$$ is the assertion that $$\{\log x_k \mid k\geq 1\}$$ is uniformly distributed $$\pmod 1$$.
This paper studies the initial iterates $$x_k=T^{(k)}(x_0)$$ for $$1\leq k\leq\mathbb{N}$$ of the $$3x+1$$ function, where $$N$$ is fixed. It is shown in the main result of this paper (Theorem 2.1) that for most initial values $$x_0$$, such sequences approximately satisfy Benford’s law, in the sense that the discrepancy of the finite sequence $$\{\log_Bx_k\mid k\geq 1\}$$ is small; there $$B\geq 2$$ is a fixed integer base.

##### MSC:
 11B83 Special sequences and polynomials 11B37 Recurrences 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
##### Keywords:
special sequences; $$3x+1$$ problem; Benford’s law
Full Text: