On Benford's Law for multiplicative functions

Abstract

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the k-divisor functions, where k = 10 j , and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.From now on, we will refer to strong Benford's law simply as "Benford's law."

Benford behavior and distribution in residue classes of large prime factors

“…For example, it is shown there that ϕ(n) is not Benford, but that |τ(n)| is, where τ is Ramanujan's τ-function. 1 All of the work in [8] is carried out in base 10, but both of the quoted results hold, by simple modifications of the proofs, in each fixed base b ≥ 2.…”

“…This criterion was noted by Aursukaree and Chandee [3] and used by them to show that the divisor function d(n) is Benford in base 10. A more systematic study of the Benford behavior of multiplicative functions, leveraging Halász's celebrated mean value theorem, was recently undertaken in [8]. For example, it is shown there that ϕ(n) is not Benford, but that |τ(n)| is, where τ is Ramanujan's τ-function.…”