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

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 $a(n) = \pi(n)$.