THE CLASS NUMBER OF Q ( √ − p ) AND DIGITS OF 1

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 1 p = ∞ ∑ k=1 xk gk 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 B1,χ (where χ is a character modulo p of order r with χ(−1) = −1). Also, we study, using Bernoulli polynomials and Dirichlet Lfunctions, the set equidistribution modulo 1 of the elements of the subgroup Hn of (Z/nZ) ∗ as n → ∞ whenever |Hn|/ √ n → ∞.