Mean values and upper bounds for the Hurwitz and Barnes multiple zeta functions

Due to their deep connection with the Riemann zeta function, the asymptotic behavior of mean values of multiple zeta functions has attracted considerable attention. In this paper, we study the mean square values of Hurwitz-type and Barnes-type multiple zeta functions. For the Hurwitz-type multiple zeta function, we establish asymptotic formulas and upper bounds for its mean square values in terms of the parameter $\sigma$. Our approach relies on the fact that Hurwitz-type multiple zeta functions can be expressed as linear combinations of the classical Hurwitz zeta function, which allows us to apply known results on the mean values of the latter almost directly. For the Barnes-type multiple zeta function, we show that the behavior of the mean square values depends essentially on the arithmetic structure of the parameter vector. In the case where the parameters are linearly dependent over $\Q$, we obtain asymptotic formulas analogous to the Hurwitz-type case. In contrast, for general parameters, we derive upper bounds for the mean square values from bounds for the function itself. In particular, we clarify how the order of the mean square values varies in terms of the dimension of the $\Q$-vector space spanned by the parameters of the Barnes zeta function.