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 establish asymptotic formulas and upper bounds for the mean square values of Hurwitz-type and Barnes-type multiple zeta functions. In particular, we focus on the Hurwitz-type case, since Hurwitz 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 and asymptotic behavior of the latter almost directly. Moreover, it can be shown that Hurwitz-type and Barnes-type multiple zeta functions have the same order under certain conditions. This fact enables us to investigate the mean values and growth of Barnes multiple zeta functions, which are otherwise difficult to evaluate, by using the results for Hurwitz multiple zeta functions.

💡 Research Summary

The paper investigates the mean‑square behavior of two families of multiple zeta functions: the Hurwitz‑type and the Barnes‑type. The Barnes multiple zeta function is defined for a positive integer r, a>0 and a vector of positive weights w=(w₁,…,wᵣ) by

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