The class number of $\mathbb{Q}(\sqrt{-p})$ and digits of $1/p$

Abstract

Let p be a prime number such that p ≡ 1 (mod r) for some integer r > 1. Let g > 1 be an integer such that g has order r in (Z/pZ) *. Let be the g-adic expansion. Our result implies that the average digit in the g-adic expansion of 1/p is (g — 1)/2 with error term involving the generalized Bernoulli numbers B 1,χ (where χ is a character modulo p of order r with χ(―1) = ―1). Also, we study, using Bernoulli polynomials and Dirichlet L-functions, the set equidistribution modulo 1 of the elements of the subgroup H n of (ℤ/nℤ) * as n → ∞ whenever |H n |/√n → ∞.