Generalization of the Von Staudt-Clausen theorem

AbstractThe localization LS(x) of log(1 + x) at a set of primes S is defined by taking those powers of x in the logarithmic series for log(1 + x) which lie in the span of S. The functional inverse LS−1(x) of LS(x) also localizes the functional inverse ex − 1 of log(1 + x) and a generalization of the Von Staudt-Clausen theorem is proved for the even coefficients in the power series expansion for xLS−1(x). This reduces to the Von Staudt-Clausen theorem when S is the set of all primes and to a weaker version of Theorem 3.9 of I. Dibag (J. Algebra87 (1984), 332–341) when S consists of a single prime