Number field analogue of divisor function \`a la Koshliakov

In this article, we study a divisor function in an arbitrary number field akin to Koshliakov’s work on Vorono"{\dotlessi} summation formula. More precisely, we generalize Koshliakov’s kernel and Koshliakov’s transform over any number field to obtain identities for the Lambert series associated to the divisor function in an arbitrary number field.