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

Takashi Miyagawa

TL;DR

The paper analyzes mean values and upper bounds for Hurwitz-type and Barnes-type multiple zeta functions. By expressing the Hurwitz-type $\zeta_r(s,a,\mathbf{1})$ as a linear combination of single Hurwitz zeta functions, it transfers known mean-value results to the multi-parameter setting and derives precise mean-square asymptotics for $\zeta_r(\sigma+it,a,\mathbf{1})$ in the strip $r-1<\sigma<r$, including distinct regimes with main and secondary terms. It further shows that Barnes-type zeta functions share the same order under certain conditions, enabling the use of Hurwitz-based results to study Barnes zeta, and provides growth bounds via a Hankel-type contour representation. The combination of diagonal/off-diagonal analysis, Montgomery–Vaughan-type bounds, and functional-analytic techniques yields a coherent framework for mean-value theory of higher-dimensional zeta functions and clarifies their analytic behavior across different regions of $\sigma$. These results contribute to a deeper understanding of the mean-value phenomena for higher-order zeta functions and their interrelations.

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 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.

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

TL;DR

The paper analyzes mean values and upper bounds for Hurwitz-type and Barnes-type multiple zeta functions. By expressing the Hurwitz-type as a linear combination of single Hurwitz zeta functions, it transfers known mean-value results to the multi-parameter setting and derives precise mean-square asymptotics for in the strip , including distinct regimes with main and secondary terms. It further shows that Barnes-type zeta functions share the same order under certain conditions, enabling the use of Hurwitz-based results to study Barnes zeta, and provides growth bounds via a Hankel-type contour representation. The combination of diagonal/off-diagonal analysis, Montgomery–Vaughan-type bounds, and functional-analytic techniques yields a coherent framework for mean-value theory of higher-dimensional zeta functions and clarifies their analytic behavior across different regions of . These results contribute to a deeper understanding of the mean-value phenomena for higher-order zeta functions and their interrelations.

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 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.
Paper Structure (4 sections, 11 theorems, 87 equations)

This paper contains 4 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction and main theorems
  2. The Hurwitz and Lerch zeta functions
  3. Some Lemmas
  4. Proofs of the Main Theorems

Key Result

Theorem 1.1

Let $s=\sigma+it$ and assume $r-1<\sigma < r$. Then the order of the mean square values of $\zeta_r(\sigma+it,a,\mathbf{1})$ is given in the following three cases: Here $\gamma$ denotes Euler's constant, and $\gamma(a)$ the generalized Euler constant, defined by

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Proposition 2.1: Theorem 2.1 and 2.2 for Chapter 4 in LaurincikasGarunkstis2002
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 8 more