Two problems on the distribution of Carmichael's lambda function

2023/03/24 by Paul Pollack, Pollack, Paul
Computer Science · Mathematics · #11B50 #Authorship Attribution and Profiling #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT) #Primary 11N64 #Secondary 11A25

paper · pdf · doi:10.48550/arxiv.2303.14043

openalex publication_date 2023/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let λ(n) denote the exponent of the multiplicative group modulo n. We show that when q is odd, each coprime residue class modulo q is hit equally often by λ(n) as n varies. Under the stronger assumption that gcd(q,6)=1, we prove that equidistribution persists throughout a Siegel--Walfisz-type range of uniformity. By similar methods we show that λ(n) obeys Benford's leading digit law with respect to natural density. Moreover, if we assume GRH, then Benford's law holds for the order of a mod n, for any fixed integer a∉ \0,± 1\.

Related

• The iterated Carmichael lambda function
• Mean values of multiplicative functions and applications to the distribution of the sum of divisors
• Joint distribution in residue classes of families of polynomially-defined multiplicative functions
• On a problem of Arnold: the average multiplicative order of a given integer
• The distribution of d₄(n) in arithmetic progressions