ℓ-Adic digits and class number of imaginary quadratic fields.

Motivated by the work of [K. Girstmair, A "popular" class number formula, Amer. Math. Monthly 101(10) (1994) 997–1001; K. Girstmair, The digits of 1 / p in connection with class number factors, Acta Arith. 67(4) (1994) 381–386] and [M. R. Murty and R. Thangadurai, The class number of ℚ (− p) and digits of 1 / p , Proc. Amer. Math. Soc. 139(4) (2010) 1277–1289], we study the average of the digits of the ℓ -adic expansion of 1 / n whenever n is a product of two distinct primes or a prime power. More explicitly, if ℓ > 1 is an integer such that gcd (ℓ , n) = 1 , and suppose that 1 / n = ∑ k = 1 ∞ x k ℓ k is the ℓ -adic expansion of 1 / n , then we establish the average of the digits of the ℓ -adic expansion of 1 / n in terms of (ℓ − 1) / 2 and the "trace" of generalized Bernoulli numbers B 1 , χ , where χ 's are odd Dirichlet characters modulo n. As a consequence of these results, we recover two well-known results of Gauss and Heilbronn (see Theorems 1.6 and 1.7). [ABSTRACT FROM AUTHOR]