2023/09/24 by John M. Campbell, Campbell, John M.
Computer Science · Mathematics · #11K16 #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2309.13520
openalex publication_date 2023/09/24 · openalex created_date 2023/09/27 · openalex updated_date 2026/07/28
Let (a(n) : n ∈ 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 ∈ 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-Erdős constant 0.2357111317..., for the case whereby a(n) equals the nth prime number pn, is normal in base 10. However, it seems that the ``inverse'' construction given by concatenating the decimal digits of (π(n) : n ∈ ℕ), where π 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 ∈ ℕ appears in (π(n) : n ∈ ℕ) is equal to the prime gap gm = pm+1 - pm, 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)... in a given base g ≥ 2, for a(n) = π(n).