The prime-counting Copeland–Erdős constant

Abstract

Let \((a(n) : n \in \mathbb{N})\) denote a sequence of nonnegative integers. Let \(0.a(1)a(2) \ldots \) 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 \(0.a(1)a(2) \ldots \) for a given base. Famously, the Copeland-Erdős constant \(0.2357111317 \ldots {}\), 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 \(0.0122 \ldots 9101011 \ldots \) 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 Szüsz and Volkmann, we prove that Cramér’s conjecture on prime gaps implies the normality of \(0.a(1)a(2) \ldots \) in a given base \(g \geq 2\), for \(a(n) = \pi(n)\).