? -Adic digits and class number of imaginary quadratic fields

Abstract

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 ?-adic expansion of 1/n whenever n is a product of two distinct primes or a prime power. More explicitly, if ? > 1 is an integer such that gcd(? n) = 1, and suppose that 1/n =?k=1?xk/? k is the ?-adic expansion of 1/n, then we establish the average of the digits of the ?-adic expansion of 1/n in terms of (? - 1)/2 and the "trace"of generalized Bernoulli numbers B1,?, where ?'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). � 2024 World Scientific Publishing Company.