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A multiple zeta value is defined as
[tex] \zeta(s_1,...,s_k) = \sum_{n_1 > n_2 ... > n_k > 0} \frac{1}{n_1^{s_1} n_2^{s_2}...n_k^{s_k}} [/tex].
For example,
[tex] \zeta(4) = \sum_{n = 1}^{\infty} \frac{1}{n^4} [/tex]
and
[tex] \zeta(2,2) = \sum_{m =1}^{\infty} \sum_{n = 1}^{m-1} \frac{1}{ m^2 n^2} [/tex].
Prove the following relationship:
[tex] \zeta(2)^2 = 4 \zeta(3,1) + 2 \zeta(2,2) [/tex]
[tex] \zeta(s_1,...,s_k) = \sum_{n_1 > n_2 ... > n_k > 0} \frac{1}{n_1^{s_1} n_2^{s_2}...n_k^{s_k}} [/tex].
For example,
[tex] \zeta(4) = \sum_{n = 1}^{\infty} \frac{1}{n^4} [/tex]
and
[tex] \zeta(2,2) = \sum_{m =1}^{\infty} \sum_{n = 1}^{m-1} \frac{1}{ m^2 n^2} [/tex].
Prove the following relationship:
[tex] \zeta(2)^2 = 4 \zeta(3,1) + 2 \zeta(2,2) [/tex]