Abstract: In this paper we introduce a symmetric zeta function, prove that it can be analytically continued to a meromorphic function on $\mathbb{C}^3$ with only simple poles on certain special hyperplanes, and calculate the multiple residue values at particular singular points. For a divergent multiple series that can be regarded as the value of the symmetric zeta function at the point $(1,1,1)$, we conduct a detailed analysis on its growth behavior and establish a connection with the classical Euler constant.

Key words: Multiple zeta value , Riemann zeta function