\zeta({{2}^m, 1, {2}^m, 3}^n, {2}^m) / \pi^{4n + 2m(2n+1)} is rational

TL;DR

This paper proves that inserting all permutations of certain blocks of 2's into a specific multiple zeta value results in a rational multiple of a power of pi, using motivic multiple zeta values.

Contribution

It establishes a symmetric insertion result for multiple zeta values, generalizing the cyclic insertion conjecture through motivic methods.

Findings

Inserting all permutations of fixed blocks of 2's yields rational multiples of powers of pi.

The result is non-explicit but confirms a symmetric version of the cyclic insertion conjecture.

Uses motivic multiple zeta values to prove the rationality of the resulting expressions.

Abstract

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that inserting all cyclic shifts of some fixed blocks of 2's into the multiple zeta value {\zeta}(1,3,...,1,3) gives an explicit rational multiple of a power of {\pi}. In this paper we use motivic multiple zeta values to establish a non-explicit symmetric insertion result: inserting all possible permutations of some fixed blocks of 2's into {\zeta}(1,3,...,1,3) gives some rational multiple of a power of {\pi}.