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## Multiple zeta values of weight =10 and depth=4

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### Multiple zeta values of weight =10 and depth=4

Wed Feb 22, 2017 12:48 pm

Posts: 105
The multiple zeta values are defined by the following series
$\zeta \left( \mathbf{S} \right) \equiv \zeta \left( {{s_1},{s_2}, \cdots ,{s_k}} \right) := \sum\limits_{{n_1} > {n_2} > \cdots > {n_k} \ge 1} {\frac{1}{{n_1^{{s_1}}n_2^{{s_2}} \cdots n_k^{{s_k}}}}} ,\tag{1.1}$
where ${s_1} + \cdots + {s_k}$ is called the weight and $k$ is the depth.
A good deal of work on Euler sums has been focused on the problem of determining when
complicated sums can be expressed in terms of simpler sums. Thus, researchers are interested in
determining which sums can be expressed in terms of other sums of lesser depth.
For example, we have
$\zeta(2,1)=\zeta(3),\zeta(4,1)=\frac{1}{4}\zeta(4).$
Question: The following multiple zeta values of weight =10 and depth=4 can be expressed in terms of zeta values of lesser depth?
$\zeta(5,3,1,1),\zeta(3,5,1,1),\zeta(6,2,1,1),\zeta(2,6,1,1).$