On a certain divisor function in Number fields

The main aim of this paper is to study an analogue of the generalized divisor function in a number field K, namely, ?K,?(n). The Dirichlet series associated to this function is ?K(s)?K(s ??). We give an expression for the Riesz sum associated to ?K,?(n), and also extend the validity of this formula by using convergence theorems. As a special case, when K = Q, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for ? = 0, gives the Vorono summation formula associated to the divisor counting function d(n). We also obtain a big O-estimate for the Riesz sum associated to ?K,?(n).