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arxiv: 2507.18152 · v2 · submitted 2025-07-24 · 🧮 math.NT

On the Laurent series expansions of the Barnes double zeta function

Takashi Miyagawa This is my paper

Pith reviewed 2026-05-19 03:03 UTC · model grok-4.3

classification 🧮 math.NT
keywords Barnes double zeta functionLaurent series expansionEuler-Stieltjes constantsdouble sumslogarithmic correctionsasymptotic behaviormeromorphic continuation
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The pith

The Barnes double zeta function has explicit Laurent expansions at s=1 and s=2 given by finite double sums with logarithmic corrections.

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

This paper derives explicit limit expressions for the coefficients in the Laurent series of the Barnes double zeta function around its poles. These coefficients serve as double-variable analogues of the Euler-Stieltjes constants and are expressed using finite double sums plus log terms. The work also examines their asymptotic behavior, which turns out simpler than the corresponding coefficients for the Hurwitz zeta function. A sympathetic reader would care because these expansions help decode the arithmetic content hidden near the singularities of multiple zeta functions.

Core claim

The central claim is that the Laurent coefficients of the Barnes double zeta-function ζ₂(s,α;v,w) at the points s=1 and s=2 admit representations as finite double sums together with logarithmic correction terms, obtained via explicit limit expressions, and that their asymptotic behavior is markedly simpler than that of the coefficients for the Hurwitz zeta-function.

What carries the argument

The explicit limit expressions that isolate the Laurent coefficients from the meromorphic continuation of the Barnes double zeta function, yielding representations in terms of finite double sums and logs.

If this is right

  • The coefficients act as analogues of the Euler-Stieltjes constants for this double setting.
  • These expressions provide a clearer view of the analytic structure near the poles.
  • The asymptotic simplicity contrasts with the more involved structure in the single-variable Hurwitz case.
  • Such representations highlight differences from classical zeta theory.

Where Pith is reading between the lines

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

  • This may enable more efficient numerical evaluation of the coefficients for specific parameters.
  • Similar techniques could extend to higher-order Barnes zeta functions or other multiple zeta variants.
  • The simpler asymptotics might imply new relations or bounds not available in the classical setting.

Load-bearing premise

The Barnes double zeta function possesses a meromorphic continuation to the complex plane featuring only simple poles at s=1 and s=2.

What would settle it

A direct numerical computation of the Barnes double zeta function near s=1 or s=2 that fails to match the values predicted by the proposed double-sum expressions with log corrections would falsify the explicit formulas.

read the original abstract

The Laurent series expansions of zeta-functions play an important role in understanding their behavior near singularities, and their coefficients often encode significant arithmetic information. In the case of the Riemann and Hurwitz zeta-functions, these coefficients are given by the Euler-Stieltjes constants and their generalizations. In this paper, we investigate the Laurent series expansions of the Barnes double zeta-function $\zeta_2(s,\alpha;v,w)$ at the singular points $s=1$ and $s=2$. We derive explicit limit expressions for the Laurent coefficients, providing analogues of the Euler-Stieltjes constants in this setting. In particular, we obtain representations of the coefficients in terms of finite double sums together with logarithmic correction terms. Furthermore, we study the asymptotic behavior of the Laurent coefficients and show that, in contrast to the Hurwitz zeta-function, they exhibit a much simpler structure. These results provide a clearer understanding of the analytic structure of the Barnes double zeta-function and highlight notable differences from the classical theory.

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.

Referee Report

0 major / 2 minor

Summary. The paper investigates the Laurent series expansions of the Barnes double zeta-function ζ₂(s,α;v,w) at its simple poles s=1 and s=2. It derives explicit limit expressions for the Laurent coefficients as analogues of the Euler-Stieltjes constants, expressed via finite double sums together with logarithmic correction terms. The work further examines the asymptotic behavior of these coefficients and contrasts their simpler structure with that of the Hurwitz zeta-function.

Significance. If the explicit representations and asymptotic comparisons hold, the results supply concrete, computable formulae for the coefficients that clarify the analytic structure of the Barnes double zeta function and underscore differences from the classical single-variable case. Such formulae could support numerical evaluations and further arithmetic investigations of multiple zeta functions.

minor comments (2)
  1. [Introduction] §1: the statement that the meromorphic continuation is 'known' would benefit from a specific reference to a standard source (e.g., the original Barnes papers or a modern treatment) to guide readers unfamiliar with the double-zeta literature.
  2. [Main results] The notation for the double sum in the limit expressions (presumably around the main theorems) should be checked for consistency with the definition of ζ₂(s,α;v,w) to avoid any ambiguity in the indices.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We appreciate the recognition that our explicit double-sum-plus-log representations for the Laurent coefficients at s=1 and s=2, together with the simpler asymptotics relative to the Hurwitz case, clarify the analytic structure of the Barnes double zeta function.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper starts from the known meromorphic continuation of the Barnes double zeta function (a standard result in the literature, independent of this work) with simple poles at s=1 and s=2. It then applies standard limit procedures to isolate Laurent coefficients, expressing them via finite double sums plus logarithmic terms. These steps mirror classical extractions of Stieltjes constants for the Riemann and Hurwitz zeta functions but are applied here without reducing to fitted parameters, self-definitions, or load-bearing self-citations. No equation in the provided abstract or description equates a claimed prediction back to its input by construction; the explicit representations add new content relative to the input analytic structure.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivations rest on the standard meromorphic continuation of the Barnes double zeta and the existence of the indicated limits; no free parameters or new entities are mentioned in the abstract.

axioms (1)
  • domain assumption Barnes double zeta admits meromorphic continuation with only simple poles at positive integers s=1,2
    Required for Laurent series to exist at those points; invoked implicitly by the abstract's focus on expansions at s=1 and s=2.

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Reference graph

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20 extracted references · 20 canonical work pages

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