# Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

## TL;DR

This paper links the asymptotic behavior of cotangent sum matrices to the generalized Riemann hypothesis (GRH) for odd Dirichlet characters and explores their connection to quantum modular forms associated with Maass Eisenstein series.

## Contribution

It establishes an equivalence between cotangent sum matrix asymptotics and the GRH, and connects this to quantum modular forms related to Maass Eisenstein series.

## Key findings

- Asymptotic properties of cotangent sum matrices are equivalent to GRH for odd Dirichlet characters.
- Connection between cotangent sums and quantum modular forms is demonstrated.
- Provides a new perspective on GRH through quantum modular forms.

## Abstract

We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence of certain quantum modular forms related to Maass Eisenstein series.