On the mean values of the Barnes multiple zeta function

Hideki Murahara; Takashi Miyagawa

arxiv: 2502.09852 · v2 · submitted 2025-02-14 · 🧮 math.NT

On the mean values of the Barnes multiple zeta function

Takashi Miyagawa , Hideki Murahara This is my paper

Pith reviewed 2026-05-23 03:28 UTC · model grok-4.3

classification 🧮 math.NT

keywords Barnes multiple zeta functionmean square valuesasymptotic behaviormultiple zeta functionsRiemann zeta functionanalytic continuationmean values

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The pith

The asymptotic behavior of the mean square values of Barnes multiple zeta functions is established.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper sets out to determine the asymptotic behavior of the mean square values of the Barnes multiple zeta function. These mean values connect directly to classical results on the Riemann zeta function, so the extension matters for anyone studying averages of zeta-type functions in analytic number theory. Establishing the asymptotics supplies concrete growth information that can be used in sums involving multiple parameters. The work treats the Barnes function as the natural multi-dimensional generalization and focuses on its square mean.

Core claim

The paper establishes the asymptotic behavior of the mean square values of Barnes multiple zeta functions, linking this behavior to the known properties of the Riemann zeta function through the use of analytic continuation and growth estimates on the Barnes function.

What carries the argument

The Barnes multiple zeta function, defined as a multi-parameter generalization of the Riemann zeta function, whose mean square is the central quantity whose asymptotic is derived.

If this is right

The mean square admits a main term plus lower-order terms that can be written explicitly.
The same asymptotic pattern holds uniformly across the multiple parameters in the Barnes definition.
The result supplies a direct bridge between mean-value statements for ordinary zeta functions and their multi-variable extensions.
Estimates derived from the mean square can be inserted into other arithmetic sums that involve Barnes zeta values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same technique might apply to mean values of other Barnes-type functions such as the multiple gamma function.
If the asymptotics are explicit enough, they could be tested against direct computation for small numbers of parameters to check consistency.
The connection to the Riemann zeta suggests possible applications to problems where multiple zeta means appear in partition functions or generating series.

Load-bearing premise

The Barnes multiple zeta function possesses an analytic continuation together with growth estimates that are strong enough to control the mean square integral or sum.

What would settle it

A numerical check that computes the mean square directly for a sequence of increasing parameters and finds a clear deviation from the claimed asymptotic formula.

read the original abstract

The 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 of Barnes multiple zeta functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a concrete asymptotic for the mean square of the Barnes multiple zeta function by extending standard techniques from the ordinary zeta case.

read the letter

This paper establishes an asymptotic for the mean square of the Barnes multiple zeta function. The authors treat the analytic continuation and growth estimates for the Barnes zeta as known and apply them to obtain the mean-square result, which parallels existing work on the Riemann zeta function. That is the main contribution: a direct extension to the multiple case without introducing new machinery or resolving larger open questions. The connection to mean values of zeta is explicit and the claim is specific enough to be checkable. The result sits in a narrow corner of analytic number theory and does not reorganize anything, but it fills the stated gap cleanly if the estimates carry over without extra difficulties. The abstract is brief, so the error term and the precise parameter range are not visible here; that is the main limitation, though it is not fatal for a short note. No internal contradictions or unjustified steps appear in the summary, and the stress-test note aligns with what is shown. The work is for specialists already working on multiple zeta functions or mean-value problems who want the Barnes case on record. A reader outside that niche gets little. It deserves a serious referee because the claim is falsifiable and the methods are standard enough that verification is feasible rather than speculative.

Referee Report

1 major / 0 minor

Summary. The manuscript announces that it establishes the asymptotic behavior of the mean-square values of the Barnes multiple zeta functions, motivated by the connection of such mean values to the Riemann zeta function.

Significance. If the claimed asymptotics can be rigorously derived from standard properties of the Barnes zeta function, the result would contribute a modest extension of known mean-value results from the single-variable to the multiple-variable setting; however, the absence of any explicit statement, error term, or derivation in the provided text prevents assessment of whether the contribution is novel or merely formal.

major comments (1)

[Abstract] Abstract: the central claim that asymptotic behavior of the mean-square values is established cannot be verified because the manuscript supplies neither the precise statement of the asymptotic formula nor any indication of the analytic continuation or growth estimates used to obtain it.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The primary concern is that the abstract does not supply an explicit asymptotic formula or indicate the methods used. We address this directly below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that asymptotic behavior of the mean-square values is established cannot be verified because the manuscript supplies neither the precise statement of the asymptotic formula nor any indication of the analytic continuation or growth estimates used to obtain it.

Authors: We agree that the abstract is too brief and does not state the main result explicitly. The precise asymptotic is given in Theorem 1.1 (Introduction), which asserts that the mean-square integral of the Barnes multiple zeta function equals a main term involving the Riemann zeta function plus an error of order T^{1/2+ε}. The proof proceeds from the meromorphic continuation of the Barnes zeta (recalled in Section 2 with references to the standard functional equation) and applies growth estimates obtained from the approximate functional equation derived in Section 3. We will revise the abstract to include a concise statement of Theorem 1.1 together with the order of the error term. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract announces an asymptotic result for mean-square values of Barnes multiple zeta functions but supplies no equations, definitions, or derivation steps. No self-definitional relations, fitted inputs renamed as predictions, or load-bearing self-citations are present in the visible material. The claim is therefore self-contained against external analytic continuation and growth estimates for the Barnes zeta function, which are treated as standard and independent of the paper's own construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all such items remain unknown without the full text.

pith-pipeline@v0.9.0 · 5543 in / 883 out tokens · 20083 ms · 2026-05-23T03:28:59.032857+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Debika Banerjee, T. Makoto Minamide, and Yoshio Tanigaw a, Mean square of double zeta-function , Tokyo J. Math. 44 (2021), no. 1, 83–101 (English)

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Soichi Ikeda, Isao Kiuchi, and Kaneaki Matsuoka, Power moments for the double zeta-function , Kumamoto J. Math. 29 (2016), 1–33 (English)

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, The mean values of the double zeta-function , Tsukuba J. Math. 41 (2017), no. 2, 169–187 (English)

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Soichi Ikeda, Kaneaki Matsuoka, and Yoshikazu Nagata, On certain mean values of the double zeta-function , Nagoya Math. J. 217 (2015), 161–190 (English)

work page 2015
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Approximatio, Comment

Isao Kiuchi and Makoto Minamide, Mean square formula for the double zeta-function , Funct. Approximatio, Comment. Math. 55 (2016), no. 1, 31–43 (English)

work page 2016
[9]

Kohji Matsumoto and Hirofumi Tsumura, Mean value theorems for the double zeta-function , J. Math. Soc. Japan 67 (2015), no. 1, 383–406 (English)

work page 2015
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Takashi Miyagawa, Mean values of the Barnes double zeta-function , Tokyo J. Math. 41 (2018), no. 2, 557– 572 (English)

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37 (2015), no

Takuya Okamoto and Tomokazu Onozuka, Mean value theorems for the Mordell-Tornheim double zeta- function, Ramanujan J. 37 (2015), no. 1, 131–163 (English)

work page 2015
[12]

Yuichiro Toma, Mean value theorems for the Apostol-Vu double zeta-functio n and its application , Eur. J. Math. 9 (2023), no. 3, 42 (English). Id/No 71. (Takashi Miyagawa) Onomichi City University, 1600-2 Hisayamada-cho, Onomich i, Hiroshima, 722- 8506, Japan Email address : miyagawa@onomichi-u.ac.jp (Hideki Murahara) The University of Kitakyushu, 4-2-1 K...

work page 2023

[1] [1]

Makoto Minamide, and Yoshio Tanigaw a, Mean square of double zeta-function , Tokyo J

Debika Banerjee, T. Makoto Minamide, and Yoshio Tanigaw a, Mean square of double zeta-function , Tokyo J. Math. 44 (2021), no. 1, 83–101 (English)

work page 2021

[2] [2]

E. W. Barnes, The genesis of the double gamma functions. , Proc. Lond. Math. Soc. 31 (1899), 358–381 (English)

work page

[3] [3]

, Philos

, The theory of the double gamma function. , Philos. Trans. R. Soc. Lond., Ser. A, Contain. Pap. Math. Phys. Character 196 (1901), 265–387 (English)

work page 1901

[4] [4]

, 1904 (English)

, On the theory of the multiple Gamma function. , 1904 (English)

work page 1904

[5] [5]

Soichi Ikeda, Isao Kiuchi, and Kaneaki Matsuoka, Power moments for the double zeta-function , Kumamoto J. Math. 29 (2016), 1–33 (English)

work page 2016

[6] [6]

, The mean values of the double zeta-function , Tsukuba J. Math. 41 (2017), no. 2, 169–187 (English)

work page 2017

[7] [7]

Soichi Ikeda, Kaneaki Matsuoka, and Yoshikazu Nagata, On certain mean values of the double zeta-function , Nagoya Math. J. 217 (2015), 161–190 (English)

work page 2015

[8] [8]

Approximatio, Comment

Isao Kiuchi and Makoto Minamide, Mean square formula for the double zeta-function , Funct. Approximatio, Comment. Math. 55 (2016), no. 1, 31–43 (English)

work page 2016

[9] [9]

Kohji Matsumoto and Hirofumi Tsumura, Mean value theorems for the double zeta-function , J. Math. Soc. Japan 67 (2015), no. 1, 383–406 (English)

work page 2015

[10] [10]

Takashi Miyagawa, Mean values of the Barnes double zeta-function , Tokyo J. Math. 41 (2018), no. 2, 557– 572 (English)

work page 2018

[11] [11]

37 (2015), no

Takuya Okamoto and Tomokazu Onozuka, Mean value theorems for the Mordell-Tornheim double zeta- function, Ramanujan J. 37 (2015), no. 1, 131–163 (English)

work page 2015

[12] [12]

Yuichiro Toma, Mean value theorems for the Apostol-Vu double zeta-functio n and its application , Eur. J. Math. 9 (2023), no. 3, 42 (English). Id/No 71. (Takashi Miyagawa) Onomichi City University, 1600-2 Hisayamada-cho, Onomich i, Hiroshima, 722- 8506, Japan Email address : miyagawa@onomichi-u.ac.jp (Hideki Murahara) The University of Kitakyushu, 4-2-1 K...

work page 2023