The normality of digits in almost constant additive functions

TL;DR

This paper investigates the digit patterns of additive functions similar to the prime counting function, establishing conditions under which concatenated digits form a normal number, thus shedding light on their randomness properties.

Contribution

It characterizes when concatenated digits of certain additive functions produce normal numbers, linking digit randomness to the parameter y and extending understanding beyond Erdős-Kac theorem limitations.

Findings

Number formed by concatenating digits is normal if and only if 0 < y ≤ 1/2.

Provides conditions for digit normality in additive functions related to prime counting.

Offers new insights into the randomness of digit patterns beyond classical probabilistic theorems.

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 \Omega(n). If we concatenate the last \lceil y \frac{\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 \le 1/2. This provides insight into the randomness of digit patterns of additive function after the Erdos-Kac theorem becomes ineffective.