# Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$

## TL;DR

This paper proves two conjectural series for the Riemann zeta function at 7, using integral representations and cyclotomic harmonic polylogarithms, and introduces methods implemented in the HarmonicSums package.

## Contribution

It provides a novel proof of two conjectured identities for (7) using a new approach with integral representations and polylogarithms, advancing the understanding of zeta series.

## Key findings

- Proved two conjectural series for (7)
- Expressed series in terms of cyclotomic harmonic polylogarithms
- Implemented methods in the HarmonicSums software

## Abstract

We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive integral representations. Using substitutions, we express these integral representations in terms of cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive the results. These methods are implemented in the computer algebra package HarmonicSums.