An analogue of the Von Staudt-Clausen theorem

Abstract

The integrality theorem, e’@’ 1 E Z,[ 1.~ ] 1, proved in [2 J and subsequently in 131, enables us to interpret L,(x) as the localization of L(X) = log( 1 + X) at the prime p. This motivates us to replace in 1 above, (eX 1) which is the functional inverse of L(X), by the functional inverse of L,(x). The purpose of this paper is to write down a power-series expansion for the function X/L;‘(X) and determine the congruences mod p satisfied by the numerators of its coeffkients. The main result in this direction is Theorem 3.9 which states that X % a,x rl(l’-II 7 L,‘(x)= n-u P I -n,ln(PI)1