The prime-counting Copeland–Erdős constant

Let \((a(n) : n \in \mathbb{N})\) $(a(n):n\in \mathbb{N})$ denote a sequence of nonnegative integers. Let \(0.a(1)a(2) \ldots \) $0.a(1)a(2)\dots$ denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of \((a(n) : n \in \mathbb{N})\) $(a(n):n\in \mathbb{N})$ . Research on digit expansions of this form has mainly to do with the normality of \(0.a(1)a(2) \ldots \) $0.a(1)a(2)\dots$ for a given base. Famously, the Copeland-Erdős constant \(0.2357111317 \ldots {}\) $0.2357111317\dots$ , for the case whereby \(a(n)\) $a(n)$ equals the \(n^{\text{th}}\) $n^{\text{th}}$ prime number \(p_{n}\) $p_n$ , is normal in base 10. However, it seems that the “inverse” construction given by concatenating the decimal digits of \((\pi(n) : n \in \mathbb{N})\) $(\pi (n):n\in \mathbb{N})$ , where \(\pi\) $\pi$ denotes the prime-counting function, has not previously been considered. Exploring the distribution of sequences of digits in this new constant \(0.0122 \ldots 9101011 \ldots \) $0.0122\dots 9101011\dots$ would be comparatively difficult, since the number of times a fixed \(m \in \mathbb{N} \) $m\in \mathbb{N}$ appears in \((\pi(n) : n \in \mathbb{N})\) $(\pi (n):n\in \mathbb{N})$ is equal to the prime gap \(g_{m} = p_{m+1} - p_{m}\) $g_m=p_{m+1}-p_m$ , 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 \(0.a(1)a(2) \ldots \) $0.a(1)a(2)\dots$ in a given base \(g \geq 2\) $g\ge 2$ , for \(a(n) = \pi(n)\) $a(n)=\pi (n)$ .