# Series representation of a cotangent sum related to the Estermann zeta function

## TL;DR

This paper derives an explicit series expansion for a cotangent sum related to the Estermann zeta function, improving upon previous identities and providing a clearer analytical representation.

## Contribution

It introduces a new explicit series formula for the cotangent sum c0(q/p) when q=1, enhancing the understanding of its relation to the Estermann zeta function.

## Key findings

- Derived an explicit series expansion for c0(1/p).
- Improved upon previous identities related to the cotangent sum.
- Provides analytical tools for further study of the Estermann zeta function.

## Abstract

In this paper, we are interested by the cotangent sum c0(q/p) related to the Estermann zeta function for the special case when q = 1 and get explicit formula for its series expansion, which represents an improvement of the identity (2:1) Theorem (2:1) in the recent work [11].