Arithmetical properties of double Möbius-Bernoulli numbers

Given positive integers n, n′ and k, we investigate the Möbius-Bernoulli numbers M _k (n), double Möbius-Bernoulli numbers M _k (n, n′), and Möbius-Bernoulli polynomials Mk(n)(x). We find new identities involving double Möbius-Bernoulli, Barnes-Bernoulli numbers and Dedekind sums. In part of this paper, the Möbius-Bernoulli polynomials Mk(n)(x), can be interpreted as critical values of the following Dirichlet type L-function (Equation presented) which has analytic continuation to the whole s-complex plane, where μ is the Möbius function.