The normality of digits in almost constant additive functions

2012/06/06 by Joseph Vandehey, Vandehey, Joseph
Computer Science · Mathematics · #11K16 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:11K16 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1206.1095

arxiv created 2012/06/06 · openalex publication_date 2012/06/06 · arxiv updated 2012/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider numbers formed by concatenating some of the base b digits from additive functions f(n) that closely resemble the prime counting function Ω(n). If we concatenate the last \lceil y (log log log n)/(log b) \rceil digits of each f(n) in succession, then the number so created will be normal if and only if 0 < y ≤ 1/2. This provides insight into the randomness of digit patterns of additive function after the Erdos-Kac theorem becomes ineffective.

Related

• Normal and pseudonormal numbers
• Normal numbers with digit dependencies
• Asymptotic normality of additive functions on polynomial sequences in canonical number systems
• The distribution of sums and products of additive functions
• From randomness in two symbols to randomness in three symbols