Number field analogue of divisor function \`a la Koshliakov

TL;DR

This paper generalizes Koshliakov's divisor function and associated transforms to arbitrary number fields, deriving identities for Lambert series in this broader context.

Contribution

It introduces a number field analogue of Koshliakov's divisor function and extends the associated summation formula and transforms to arbitrary number fields.

Findings

Derived identities for Lambert series over number fields

Generalized Koshliakov's kernel and transform

Extended Voronoi summation formula to number fields

Abstract

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.