Remark on a symmetric zeta function

. In this paper we define a symmetric zeta function. We show that it can be analytically continued to a meromorphic function on C 3 with only simple poles at some special hyperplanes. We also calculate the value of a multiple residue at one special point. For a divergent multiple series, which can be viewed as the value of the symmetric zeta function at the point (1 , 1 , 1), we give a very detailed analysis on its growth and we relate it to the classical Euler constant.