Densities of Sets of Primes Related to Decimal Expansion of Rational Numbers

Abstract

For a prime p and a given integer a ≠ 0, ±1, not divisible by p, we suppose the order of a mod p is composite nk, n > 1. Let r_i be the least non-negative residue mod p of a^ki, 0 ≤ i ≤ n — 1 (which has a relationship to a-adic expansion of 1/p), and let s_a(p) = 1/pΣ ⁿ⁻¹_i=0 r_i = Σ ⁿ⁻¹_i=0 {a^ki/p}, with {α} denoting the fractional part of α. We are interested in the set of primes for which s_a(p) takes a prescribed value s for a given n, and in particular in the limit of the relative frequency (density) P_a(x,n,s). We propose several new conjectures on them and also on P _g ^r(x, n, s) constructed in the same way from the subgroup G_n,p of (ℤ/pℤ)^x of order n, where a is taken to be a primitive root mod p. We also refer to rather general probabilistic phenomena.