On Bernoulli identities and applications (Q1265424)

The authors introduce a general principle for producing Bernoulli identities and apply it to various zeta functions. Their propositions include some of the classical results of Euler and Ramanujan. The basic identity is as follows: Let \(m_1,\ldots,m_r\) be positive integers and \(P(T)\) be a polynomial in \(T\) with complex coefficients of degree less than \(m_1+\cdots+m_r\). For \(|T|<1\), we let \[ F(T):=\frac{P(T)}{(1-T^{m_1})\cdots(1-T^{m_r})}=\sum_{k=0}^\infty a(k)T^k. \] The underlying principle is to evaluate \(F(T)\) in two ways using the associated Dirichlet series \[ Z_F(s):=\sum_{k=1}^\infty a(k) k^{-s}. \] Applying this method the first author gives, besides other results, new proofs of some classical theorems such as that by von Staudt-Clausen or Kummer congruences or \(p\)-adic interpolation.