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Office_Shredder

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## Main Question or Discussion Point

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]