On a conjecture for the dimension of the space of the multiple zeta values

Since Euler, values of various zeta functions have long attracted a lot
of mathematicians. In computer algebra community, Ap´ery’s proof of the irrationality of ζ(3) is well known. In this paper, we are concerned with the “multiple zeta value (MZV)”. More than fifteen years ago, D. Zagier gave a conjecture on MZVs based on numerical computations on PARI. Since then there have been various derived conjectures and two kinds of efforts for attacking them: one is a mathematical proof and another one is a computational experiment to get more confidence to verify a conjecture. We have checked one of these conjectures up to weight k = 20, which will be explained later, with Risa/Asir function for non-commutative polynomials and special parallel programs of linear algebra designed for this purpose.