the multi-variable analytic function, called the [**multiple zeta function**]{}, $$\zeta(s_1,\cdots, s_k)=\sum_{n_1>\cdots>n_k>0}
    \frac{1}{n_1^{s_1}\cdots n_k^{s_k}}
\mlabel{eq:mzv}$$ at positive integers $s_1,\cdots,s_k$ with $s_1>1$. With the earliest study of MZVs went back to Euler when $k=2$, their systematic study started in early 1990s with the works of Hoffman  and Zagier . Since then MZVs and their generalizations have been studied extensively by numerous authors from different point of views with connections to arithmetic geometry, mathematical physics, quantum groups and knot theory .

In comparison, little is known about special values of multiple zeta functions at integers that are not all positive. Through the recent work of Zhao  and Akiyama-Egami-Tanigawa  (see also ), we know that $\zeta(s_1,\cdots,s_k)$ can be meromorphically continued to $\CC^k$ with singularities on the subvarieties $$s_1=1; \
s_1+s_2=2,1,0,-2,-4, \cdots; {\rm\ and\ } \mlabel{eq:pole}
\sum_{i=1}^j
s_{i} \in \ZZ_{\leq j}\ (3\leq j\leq k).$$ Thus $\zeta(s_1,\cdots,s_k)$ is undefined at most points with non-positive arguments. In , several definitions were proposed for the non-positive MZVs, that is, the values of $\zeta(s_1,\cdots,s_k)$ when $s_i$ are all non-positive. Some of them are $$\lim_{r_1\to s_1}\cdots\lim_{r_k\to s_k} \zeta(r_1,\cdots,r_k),
\lim_{r_k\to s_k}\cdots\lim_{r_1\to s_1} \zeta(r_1,\cdots,r_k),
\lim_{r\to 0} \zeta(s_1+r,\cdots,s_k+r).$$ As expected they give different values. Some good properties of the variously defined non-positive MZVs 