Benford’s law for linear recurrence sequences. (English) Zbl 0636.10007

Nagasaka, Kenji; Shiue, Jau-Shyong

Tsukuba J. Math. 11, 341-351 (1987).

In section 2 the authors prove: Theorem 2.1. Let \(\{y_ n\}_{n=1,2,...}\) be an integer sequence generated by the recursion formula \(y_{n+1}=ry_ n+f(n)\), \(n=1,2,..\). If the series \(\sum^{\infty}_{n=1}f(n)/r^{n-1}\) is convergent, then the sequence \(\{y_ n\}_{n=1,2,...}\) obeys Benford’s law except for the case \(r=10^ m\) with m being some nonnegativenumber of primitive representations of a positive definite quadratic form by a given even unimodular positive definite quadratic form. Finally the well-known estimate \(a_ f(T)=O((\det T)^{k/2})\) for a cusp form f is improved.

Reviewer: A.Krieg

MSC:

11B37	Recurrences
11K16	Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Keywords:

Benford’s law; number of primitive representations; quadratic form; cusp form