# Cyclotomic p-adic multi-zeta values

## TL;DR

This paper computes explicit formulas for cyclotomic p-adic multi-zeta values at all depths, advancing understanding of their properties and potential applications in number theory and p-adic Galois representations.

## Contribution

It generalizes previous results by providing explicit formulas for all depths of cyclotomic p-adic multi-zeta values.

## Key findings

- Explicit formulas for all depths of cyclotomic p-adic multi-zeta values.
- Potential applications in proving non-vanishing and transcendence.
- Implications for p-adic Galois representations.

## Abstract

The cyclotomic $p$-adic multi-zeta values are the $p$-adic periods of $\pi_{1}(\mathbb{G}_{m} \setminus \mu_{M},\cdot),$ the unipotent fundamental group of the multiplicative group minus the $M$-th roots of unity. In this paper, we compute the cyclotomic $p$-adic multi-zeta values at all depths. This paper generalizes the results in [6] and [7]. Since the main result gives explicit formulas we expect it to be useful in proving non-vanishing and transcendence results for these $p$-adic periods and also, through the use of $p$-adic Hodge theory, in proving non-triviality results for the corresponding $p$-adic Galois representations.