Generalization of the von Staudt-Clausen theorem (Q1824648)

For a set of primes S the author defines the ``span'' \(Q_ S\) by \(Q_ S=\{n\in {\mathbb{Z}}:\) \(n=p_ 1^{a_ 1}... p_ k^{a_ k}\), \(p_ i\in S\), \(a_ i\in {\mathbb{Z}}\), \(a_ i\geq 0\}\) and the ``localization'' \(L_ S(x)\) of \(\log (1+x)\) by \(L_ S(x)=\sum_{n}((-1)^{n-1}/n)x^ n\quad (n\in Q_ S).\) Let \(L_ s^{-1}(x)\) be the functional inverse of \(L_ S(x)\) and let \(b_ n\) be rational numbers satisfying the equality \(x/L_ S^{-1}(x)=\sum^{\infty}_{n=0}b_ n(x^ n/n!).\) The main result of this paper is Theorem 1.4: \[ b_{2n}=-\sum_{p- 1/2n,\quad p\in S}1/p\quad (mod {\mathbb{Z}}). \] If S is the set of all primes, then we obtain the von Staudt-Clausen Theorem \((b_ n=B_ n\) is the Bernoulli number). If \(S=\{p\}\) is a one-element set, then we get a former theorem of the author [J. Algebra 87, 332-341, Theorem 3.9 (1984; Zbl 0536.10012)].