Non-vanishing of certain cyclotomic multiple harmonic sums and application to the non-vanishing of certain $p$-adic cyclotomic multiple zeta values

TL;DR

This paper establishes the non-vanishing of specific cyclotomic multiple harmonic sums and uses this to prove the non-vanishing of certain $p$-adic cyclotomic multiple zeta values, advancing understanding in $p$-adic number theory.

Contribution

It introduces a method to prove non-vanishing of cyclotomic sums and applies it to $p$-adic zeta values, providing new non-vanishing results.

Findings

Proved non-vanishing of certain cyclotomic multiple harmonic sums.

Established non-vanishing of specific $p$-adic cyclotomic multiple zeta values.

Connected harmonic sums to zeta values via a new formula.

Abstract

We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.