Let l be a positive integer and k_1, …, k_l be integers such that k_i ≥ 1 for i = 1, …, l-1 and k_1 ≥ 2. We define the multiple zeta value ξ(k_1, …, k_l) of index (k_1, …, k_l) as ξ(k_1, …, k_l) = Σ from m_1 < … < m_l of 1/(m_1~(k_1) … m_l~(k_l). Set h_0 = 0 and h_i = Σ_(j=1)~ik_i. For an index (k_1, …, k_l), the number n = Σ_(i=1)~lk_i is called the weight of the index. The weight of the multiple zeta value ξ(k_1, …, k_l) is defined to be n. We have the following integral expression of ξ(k_1, …, k_l): ξ(k_1, …, k_l) = (-1)~l ∫ from x = △n Π_(i=1)~l[(dx_(h_i)+1)/(x_(h_(i-1)+1)-1) ∧ Π_(j=1)~(k_i-1) (dx_(h_(i-1))+j+1)/x_(h_(i-1)+j+1)] Here △_n is the topological cycle defined by {0 < x_1 < … < x_n < 1}. We define the subspace L_n of R by L_n = Σ from k_1 + … + k_l = n, k_i ≥ 1, k_l ≥ 2 of ξ(k_1, …, k_l)Q.
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