On the estimation of the order of Euler-Zagier multiple zeta-functions (Q1419641)

In this paper an estimate of the Euler-Zagier multiple zeta function is given: \[ \zeta_r(s_1,\ldots,s_r)=\sum_{n_1=1}^\infty\cdots \sum_{n_r=1}^\infty n_1^{-s_1}(n_1+n_2)^{-s_2}\cdots (n_1+\cdots+n_r)^{-s_r}. \] To this end, authors use a Mellin-Barnes integral formula: \[ \zeta_r(s_1,\ldots,s_r)=\frac1{2\pi i}\int_{(c)} \frac{\Gamma(s_r+z)\Gamma(-z)}{\Gamma(s_r)}\zeta_{r-1}(s_1,\ldots,s_{r-2},s_{r-1}+s_r +z)\zeta(-z)\,dz, \] where \(\zeta\) is the Riemann zeta-function and \((c)\) is an infinite line from \(c-i\infty\) to \(c+i\infty\) (\(c\) is a real number). Shifting the path to the right, the desired estimate is given. When \(r=2\) or \(r=3\), more improved estimates are shown by shifting the path to the left.