On the nature of Benfordʹs Law

We study multiplicative and affine sequences of real numbers defined by N(j+1)=ζ(j)N(j)+η(j) ,where {ζ(j)} and {η(j)} are sequences of positive real numbers (in the multiplicative case η(j)=0 for all j). We investigate the conditions under which the leading digits k of {N(j)} have the following probability distribution, known as Benfordʹs Law, P(k)=log10((k+1)/k). We present two main results. First, we show that contrary to the usual assumption in the literature, {ζ(j)} does not necessarily need to come from a chaotic or independent random process for Benfordʹs Law to hold. The multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing. Second, we give conditions under which the distribution of the first digits of an affine process displays Benfordʹs Law. Our proofs use techniques from ergodic theory.