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arxiv: 2606.09544 · v1 · pith:7JWB36WRnew · submitted 2026-06-08 · 🧮 math.NT

Multiple zeta values ending with a fixed string

Steven Charlton This is my paper

Pith reviewed 2026-06-27 14:54 UTC · model grok-4.3

classification 🧮 math.NT
keywords multiple zeta valuesgenerating seriesOhno-Zagier theorembounded depthinterpolated zeta valuesdouble zeta valuesheight
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The pith

A generating series sums multiple zeta values of fixed weight and depth ending with a given string, showing the total has depth bounded by the sum of the string entries.

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

The paper constructs a generating series that collects all multiple zeta values of given weight, depth, and height terminating in a prescribed index string. This extends the method used to prove the Ohno-Zagier theorem on sums of MZVs. The resulting expression immediately implies that any such sum over values ending with a fixed string has depth at most the sum of the entries in that string. The bound supplies new information on the structure of MZVs and yields explicit applications to interpolated zeta values and to the generating series for double zeta values.

Core claim

We give a generating series expression for the sum of all multiple zeta values of a fixed weight, depth, and height, which end with a given string vec ell = (ell_1, ..., ell_r); this builds upon the proof of the Ohno-Zagier Theorem. In particular, the sum of all multiple zeta values of fixed weight, depth and ending with vec ell has bounded depth ≤ ell_1 + ⋯ + ell_r. We give some applications to evaluations of interpolated multiple zeta values, and to the generating series of double zeta values.

What carries the argument

The generating series for MZVs of fixed weight, depth and height ending with a prescribed string vec ell, constructed by extending the Ohno-Zagier proof technique.

If this is right

  • Sums of MZVs of fixed weight and depth ending in a fixed string have depth at most the sum of the string entries.
  • The same generating series yields evaluations for certain interpolated multiple zeta values.
  • The construction produces the generating series for all double zeta values as a special case.

Where Pith is reading between the lines

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

  • The bound by ending-string sum may organize MZVs into families whose relations become visible by comparing different terminal strings.
  • The approach could be tested by computing low-weight examples and checking whether the depth bound is attained.
  • If the series can be evaluated at specific points it may recover known basis conjectures for MZVs of given depth.

Load-bearing premise

The extension of the Ohno-Zagier proof technique produces a well-defined generating series without new obstructions or divergences.

What would settle it

An explicit closed-form evaluation, for weight 6 and depth 3 ending in the string (2), whose depth exceeds 2 would contradict the claimed bound.

read the original abstract

We give a generating series expression for the sum of all multiple zeta values of a fixed weight, depth, and height, which end with a given string $\vec{\ell} = (\ell_1,\ldots,\ell_r)$; this builds upon the proof of the Ohno-Zagier Theorem. In particular, the sum of all multiple zeta values of fixed weight, depth and ending with $\vec{\ell}$ has bounded depth $\leq \ell_1 + \cdots + \ell_r$. We give some applications to evaluations of interpolated multiple zeta values, and to the generating series of double zeta values.

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 manuscript constructs a generating series for the sum of multiple zeta values (MZVs) of fixed weight, depth and height that terminate with a prescribed string →ℓ = (ℓ_{1}, …, ℓᵣ). The construction adapts the proof of the Ohno–Zagier theorem. As a corollary, the sum of all MZVs of fixed weight and depth ending in →ℓ lies in the Q-span of MZVs of depth at most |→ℓ|. Applications to interpolated MZVs and the generating series of double zeta values are indicated.

Significance. If the generating-series identity holds, the result supplies a concrete mechanism for producing depth bounds on partial sums of MZVs and for evaluating certain interpolated series. The direct adaptation of the Ohno–Zagier argument keeps the proof within the existing algebraic framework of MZVs and may therefore be useful for further structural investigations of the MZV algebra.

minor comments (3)
  1. [Abstract] The abstract refers to 'height' without a definition; the manuscript should state explicitly whether height is the standard notion (number of indices >1) or a modified quantity, and confirm that the generating series respects this definition throughout.
  2. The statement that the sum 'has bounded depth ≤ ℓ_{1} + ⋯ + ℓᵣ' should be accompanied by a precise reference to the theorem or corollary number where the depth bound is proved, rather than left as an immediate consequence of the generating series.
  3. Notation for the string →ℓ and the associated generating series should be introduced once in a dedicated notation subsection or at the beginning of §2 to avoid repeated re-definition in later applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. The report lists no specific major comments.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs a generating series for sums of MZVs of fixed weight, depth, and height ending in a prescribed string by adapting the known proof of the external Ohno-Zagier theorem. This is an extension and application, not a reduction of the claimed result to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The central claim (bounded depth ≤ |ℓ|) follows from the generating-function construction without internal redefinition or renaming of known results as new derivations. The cited theorem is independent prior work, satisfying the criteria for non-circular external support.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities used in the derivation.

pith-pipeline@v0.9.1-grok · 5612 in / 931 out tokens · 19468 ms · 2026-06-27T14:54:16.802338+00:00 · methodology

discussion (0)

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Reference graph

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