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

TL;DR

This paper investigates the asymptotic behavior and bounds of mean square values for Hurwitz-type and Barnes-type multiple zeta functions, revealing dependence on parameter structures and extending classical results.

Contribution

It provides new asymptotic formulas and bounds for the mean square values of these functions, especially highlighting the influence of parameter arithmetic structure.

Findings

Established asymptotic formulas for Hurwitz-type multiple zeta functions.

Derived upper bounds for Barnes-type multiple zeta functions based on parameter dependence.

Clarified how the mean square order varies with the dimension of the parameter space.

Abstract

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.