Abstract 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 M k ( 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 M k ( n )( x ), can be interpreted as critical values of the following Dirichlet type L -function $$\begin{array}{} \displaystyle L_{HM}(s;n,x):=\sum_{d|n} \sum_{m= 0}^\infty \frac{\mu(d)}{(md+x)^s} \, \, \text{(for Re} (s) \gt 1), \end{array} $$ which has analytic continuation to the whole s -complex plane, where μ is the Möbius function.