Analogues of the Lindel\"of Hypothesis for the Barnes multiple zeta function and related problems

For the Lindel\"of Hypothesis concerning the Riemann zeta function $\zeta(s)$, upper bounds as $\Im(s)\to\infty$ have been extensively studied for many years. In particular, the Lindel\"of Hypothesis is one of the most important open problems in analytic number theory. It is also known to be equivalent to certain mean value estimates, which provide a fundamental connection between pointwise upper bounds and integral mean values of zeta-functions. In this paper, we consider an analogue of the Lindel\"of Hypothesis for the Barnes multiple zeta function $\zeta_r (s,a,(w_1,\dots,w_r)) = \sum_{m_1=0}^\infty \cdots \sum_{m_r=0}^\infty (a+m_1 w_1+\cdots+m_r w_r)^{-s} $, and establish equivalent conditions in terms of integral mean values. In particular, the situation depends essentially on the $\Q$-rank of $\langle w_1,\dots,w_r\rangle$, and it is especially interesting that phenomena peculiar to the Barnes multiple zeta function appear according to this rank.