The prime-counting Copeland-Erd\H{o}s constant

TL;DR

This paper investigates the digit distribution of a new prime-related constant formed by concatenating prime-counting function values, and proves its normality under Cramér's conjecture on prime gaps.

Contribution

It introduces a novel constant based on the prime-counting function and establishes its normality assuming Cramér's conjecture, linking prime gap behavior to digit distribution.

Findings

Proves the normality of the constant assuming Cramér's conjecture.

Connects prime gap conjectures to digit distribution in constructed constants.

Uses combinatorial methods by Sz"usz and Volkmann.

Abstract

Let $(a(n) : n \in \mathbb{N})$ denote a sequence of nonnegative integers. Let $0.a(1)a(2)...$ denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of $(a(n) : n \in \mathbb{N})$. Research on digit expansions of this form has mainly to do with the normality of $0.a(1)a(2)...$ for a given base. Famously, the Copeland-Erd\H{o}s constant $0.2357111317...$, for the case whereby $a(n)$ equals the $n^{\text{th}}$ prime number $p_{n}$, is normal in base 10. However, it seems that the ``inverse'' construction given by concatenating the decimal digits of $(\pi(n) : n \in \mathbb{N})$, where $\pi$ denotes the prime-counting function, has not previously been considered. Exploring the distribution of sequences of digits in this new constant $0.0122...9101011...$ would be comparatively difficult, since the number of times a fixed $m \in \mathbb{N}$ appears in $(\pi(n) : n \in \mathbb{N})$ is equal to the prime gap $g_{m} = p_{m+1} - p_{m}$, with the behaviour of prime gaps notoriously elusive. Using a combinatorial method due to Sz\"usz and Volkmann, we prove that Cram\'er's conjecture on prime gaps implies the normality of $0.a(1)a(2)...$ in a given base $g \geq 2$, for $a(n) = \pi(n)$.