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arxiv: 2208.10786 · v3 · submitted 2022-08-23 · 🧮 math.NT

Functional equation, upper bounds and analogue of Lindel\"of hypothesis for the Barnes double zeta function

Takashi Miyagawa This is my paper

Pith reviewed 2026-05-24 11:32 UTC · model grok-4.3

classification 🧮 math.NT
keywords Barnes double zeta functionfunctional equationupper boundsLindelöf hypothesisPhragmén-Lindelöf principledouble zeta function
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The pith

The Barnes double zeta function satisfies a functional equation that implies upper bounds on its vertical growth and an analogue of the Lindelöf hypothesis.

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

The paper derives a functional equation for the Barnes double zeta function defined by the double sum over (α + v m + w n) to the power -s. It applies the Phragmén-Lindelöf convexity principle to this equation to obtain upper bounds on the size of ζ₂(σ + it, α; v, w) as the imaginary part t tends to infinity for fixed real part σ. These bounds are presented along with an analogue of the Lindelöf hypothesis for this function. A sympathetic reader would care because the same pattern of functional equation plus convexity controls the growth of the ordinary Riemann zeta function and thereby shapes its zero distribution.

Core claim

We consider the functional equations for the Barnes double zeta-function ζ₂(s, α; v, w) = ∑_{m=0}^∞ ∑_{n=0}^∞ (α + v m + w n)^{-s}. By applying this functional equation and the Phragmén-Lindelöf convexity principle, we obtain some upper bounds for ζ₂(σ + it, α; v, w) with respect to t as t → ∞, along with an analogue of the Lindelöf hypothesis.

What carries the argument

The functional equation for ζ₂(s, α; v, w), which relates values of the double sum in complementary regions of the complex plane and enables growth estimates via convexity.

Load-bearing premise

The functional equation holds for the parameters under consideration and the double zeta function satisfies the polynomial growth conditions required to apply the Phragmén-Lindelöf principle.

What would settle it

Explicit computation of ζ₂(σ + it, α; v, w) for a sequence of large t with fixed σ, α, v, w showing growth strictly larger than the derived upper bound.

read the original abstract

The functional equations of the Riemann zeta function, the Hurwitz zeta function, and the Lerch zeta function have been well known for a long time, and there is great importance in studying these zeta functions. For example, fundamental properties such as the upper bounds, the distribution of zeros, and the zero-free regions in the Riemann zeta function derive from functional equations. In this paper, we consider the functional equations for the Barnes double zeta-function $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $. Additionally, by applying this functional equation and the Phragm\'en-Lindel\"of convexity principle, we obtain some upper bounds for $ \zeta_2(\sigma + it, \alpha ; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.

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 / 3 minor

Summary. The paper derives a functional equation for the Barnes double zeta function ζ₂(s, α; v, w) = ∑_{m=0}^∞ ∑_{n=0}^∞ (α + v m + w n)^{-s} via contour integration or Mellin-Barnes representation. It then applies the Phragmén-Lindelöf convexity principle to this functional equation to obtain upper bounds on |ζ₂(σ + it, α; v, w)| as |t| → ∞ and an analogue of the Lindelöf hypothesis.

Significance. If the derivation holds, the work extends classical results on functional equations and convexity bounds from the Riemann, Hurwitz, and Lerch zeta functions to the double zeta case. Such bounds are load-bearing for studying zero distributions and arithmetic properties of multiple zeta functions; the standard methods employed (contour integration followed by Phragmén-Lindelöf) are appropriate once parameter restrictions are fixed.

minor comments (3)
  1. The statement of the functional equation should explicitly list the precise region of validity in the s-plane and the admissible ranges for the parameters α, v, w (e.g., v, w > 0 and α in the appropriate cone) to avoid ambiguity when applying the Phragmén-Lindelöf principle.
  2. Clarify the growth estimates used to justify the application of Phragmén-Lindelöf in the relevant vertical strips; a brief remark on the polynomial growth in |t| would strengthen the transition from the functional equation to the convexity bounds.
  3. Add a short comparison with existing functional equations for related multiple zeta functions (e.g., references to works on the double zeta or Barnes zeta) to situate the novelty of the derived equation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the functional equation for the Barnes double zeta function via standard contour integration or Mellin-Barnes methods applied to the series definition under the given parameter restrictions, then invokes the external Phragmén-Lindelöf convexity principle on vertical strips to obtain growth bounds. Neither step reduces to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The Lindelöf analogue follows directly from the convexity bound without additional internal assumptions that loop back to the result itself. The approach is routine for multiple zeta functions and remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only; no explicit free parameters, invented entities, or ad-hoc axioms are stated. Relies on background analytic number theory tools.

axioms (2)
  • standard math Standard analytic continuation and functional equation techniques for zeta-type functions apply to the double sum definition.
    Implicitly invoked to obtain the functional equation from the series definition.
  • domain assumption The Phragmén-Lindelöf convexity principle applies under the growth conditions satisfied by the double zeta after analytic continuation.
    Used to convert the functional equation into explicit upper bounds.

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Works this paper leans on

15 extracted references · 15 canonical work pages

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