On the mean values of the Barnes multiple zeta function

TL;DR

This work analyzes the mean-square behavior of the Barnes multiple zeta function $\zeta_r(s,a,\mathbf{w})$, establishing asymptotic mean-square formulas with a main term $\tilde{\zeta_{r}}(\sigma,a,\mathbf{w})\,T$ and explicit error terms that depend on the real part $\sigma$ of the complex argument. The authors develop a truncated-sum/boundary-term framework via Euler–Maclaurin summation to represent $\zeta_r(s,a,\mathbf{w})$ as a finite sum plus boundary corrections, enabling precise mean-value calculations. By extending mean-value results known for double zeta and related zeta functions to the Barnes setting and handling $r=2$ as a consistency check (recovering Miyagawa’s results), the paper provides a robust toolkit for understanding the distributional properties of Barnes zeta-values and their connections to the Riemann zeta function. These results have potential applications in analytic number theory and the study of zeta-type mean values, offering explicit asymptotic regimes that clarify the growth and fluctuation behavior of Barnes zeta-values along vertical lines.

Abstract

Asymptotic behavior of the mean values of multiple zeta functions is of significant interest due to its close connection with the Riemann zeta function. In this paper, we establish asymptotic behavior of the mean square values for the Barnes multiple zeta function.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3: Theorem 3 in Miyagawa2018
• Theorem 1.4
• Remark 1.5
• proof : Proof of Theorem \ref{['main1']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof