On a certain divisor function in Number fields

Abstract

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