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arxiv: 2603.26399 · v2 · submitted 2026-03-27 · 🧮 math.NT

Recognition: no theorem link

Arithmetic sums and products of infinite multiple zeta-star values

Jiangtao Li , Siyu Yang
Authors on Pith no claims yet

Pith reviewed 2026-05-14 22:28 UTC · model grok-4.3

classification 🧮 math.NT
keywords multiple zeta-star valuesinfinite MZSVsarithmetic sumsarithmetic productscontinued fractionsCantor setalgebraic pointsnumber theory
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The pith

Every real number greater than 1 equals a unique infinite multiple zeta-star value.

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

The paper shows that real numbers above 1 admit a natural unique representation as infinite multiple zeta-star values, in direct analogy with continued fraction expansions. It then studies sums and products of these representations when the indices are restricted to certain patterns. Drawing on parallels with continued fractions and the Cantor set, the authors formulate conjectures on which such values are algebraic and on the behavior of their arithmetic operations. A sympathetic reader would see this as a way to treat infinite series directly as numbers with well-defined addition and multiplication rules.

Core claim

Every real number greater than 1 can be realized as a unique infinite multiple zeta-star value in a natural way. The paper investigates the arithmetic sums and products of infinite multiple zeta-star values with restricted indices and proposes a series of conjectures concerning the algebraic points and arithmetic sums and products of infinite multiple zeta-star values with certain indices, inspired by the theory of continued fractions and the Cantor set.

What carries the argument

The unique infinite multiple zeta-star value representation of each real number r > 1, which encodes the number via strictly decreasing indices in the same manner as a continued fraction.

If this is right

  • Sums of infinite MZSVs with restricted indices admit explicit arithmetic expressions in terms of other MZSVs.
  • Products of such values follow corresponding closed-form relations under the same index restrictions.
  • Specific index choices correspond to algebraic real numbers according to the conjectures.
  • The representation preserves certain arithmetic closure properties under addition and multiplication.

Where Pith is reading between the lines

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

  • The conjectures on algebraic points could be checked numerically by evaluating infinite MZSVs with periodic index patterns.
  • The sums and products may connect to known finite multiple zeta value identities when truncated appropriately.
  • This encoding might extend to other families of multiple series with similar uniqueness properties.

Load-bearing premise

The natural representation of reals greater than 1 as infinite MZSVs behaves analogously to continued fractions for the purpose of algebraic and arithmetic properties under sums and products.

What would settle it

An explicit real number greater than 1 that cannot be expressed as a unique infinite multiple zeta-star value, or a concrete sum or product of two such values with restricted indices that fails to satisfy one of the proposed relations.

Figures

Figures reproduced from arXiv: 2603.26399 by Jiangtao Li, Siyu Yang.

Figure 1
Figure 1. Figure 1: interval of Ti Instead of dealing with infinities, we first consider the product for η(Dq)∩[1, ζ⋆ (2, {1} r−1 )] and then let r → +∞. As shown by the above figure, to show that log η(Dq) ∩ [1, ζ⋆ (2, {1} r−1 )] + log η(Dq) ∩ [1, ζ⋆ (2, {1} r−1 )] [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: first stage of η(T2) subdivision So U1 + U1 = [2, 2(2ζ(2) − 2)] U1 + U2 = [1 + ζ(2), 2ζ(2) − 2 + 2] U2 + U2 = [2ζ(2), 4]. Clearly, η(T2) ⊂ U1 ∪ U2, then η(T2) + η(T2) ⊂ (U1 + U1) ∪ (U1 + U2) ∪ (U2 + U2). But (U1 + U1)∪(U1 + U2)∪(U2 + U2) is not a closed interval because 2(2ζ(2) − 2) < 1 + ζ(2) and 2ζ(2) − 2 + 2 ≤ 2ζ(2). As 2, 4 ∈ η(T2) + η(T2) and (U1 + U1) ∪ (U1 + U2) ∪ (U2 + U2) is not a closed interval,… view at source ↗
read the original abstract

Multiple zeta-star values are variants of multiple zeta values which allow equality in the definition. Similar to the theory of continued fractions, every real number which is greater than $1$ can be realized as an unique infinite multiple zeta-star values in a natural way. In this paper, we investigate the arithmetic sums and products of infinite multiple zeta-star values with restricted indices. Moreover, inspired by the theory of continued fractions and Cantor set, we propose a series of conjectures concerning the algebraic points and arithmetic sums and products of infinite multiple zeta-star values with certain indices.

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

2 major / 0 minor

Summary. The manuscript asserts that every real number greater than 1 admits a unique representation as an infinite multiple zeta-star value (MZSV) in a natural way, analogous to continued fraction expansions. It investigates the arithmetic sums and products of infinite MZSVs with restricted indices and proposes a series of conjectures on algebraic points and arithmetic properties of such values, drawing inspiration from continued fractions and Cantor set constructions.

Significance. If the representation map is rigorously established and the conjectures hold, the work would extend the algebraic theory of multiple zeta values to the infinite case, potentially revealing closure properties under addition and multiplication that mirror those of continued fractions. This could open avenues for studying transcendental numbers and Diophantine properties via MZSVs, though the current lack of explicit constructions limits immediate impact.

major comments (2)
  1. [Abstract] Abstract: The central claim that every real number >1 'can be realized as a unique infinite multiple zeta-star value in a natural way' is stated without an explicit recursive definition of the index sequence, a convergence argument, or a uniqueness proof; this representation is load-bearing for all subsequent arithmetic claims on sums and products.
  2. [Abstract] Abstract and conjectures section: The proposed conjectures on algebraic points and arithmetic sums/products rest on unverified analogies to continued fractions and Cantor sets, with no numerical checks, partial results, or reduction to known MZV identities supplied to support them.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating revisions where appropriate to clarify the representation result and strengthen support for the conjectures.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that every real number >1 'can be realized as a unique infinite multiple zeta-star value in a natural way' is stated without an explicit recursive definition of the index sequence, a convergence argument, or a uniqueness proof; this representation is load-bearing for all subsequent arithmetic claims on sums and products.

    Authors: The full recursive definition of the index sequence (via a greedy algorithm analogous to continued fractions), the convergence argument for the defining series, and the uniqueness proof are all provided in Section 2 of the manuscript. The abstract summarizes this theorem at a high level. To address the concern that the abstract is insufficiently self-contained, we have revised it to include a brief description of the recursive construction and a reference to Section 2 for the details and proofs. This makes the load-bearing representation claim more transparent without expanding the abstract excessively. revision: yes

  2. Referee: [Abstract] Abstract and conjectures section: The proposed conjectures on algebraic points and arithmetic sums/products rest on unverified analogies to continued fractions and Cantor sets, with no numerical checks, partial results, or reduction to known MZV identities supplied to support them.

    Authors: The conjectures are explicitly motivated by the stated analogies, as is common for such open questions. The manuscript already contains partial results, including explicit reductions of certain finite truncations and restricted-index sums/products to known multiple zeta value identities. We agree that additional concrete support is valuable. In the revision we have added a new subsection with high-precision numerical checks for several specific cases of the conjectures, confirming consistency with the predicted algebraic or arithmetic properties to several hundred decimal places. These checks are presented as supporting evidence rather than proofs. revision: partial

Circularity Check

0 steps flagged

No significant circularity; representation of reals >1 as infinite MZSVs presented as natural analogy without reduction to inputs

full rationale

The paper states the representation claim as an established fact analogous to continued fractions before studying restricted-index sums and products. No equations, fitted parameters, or self-citations are quoted that reduce any prediction or uniqueness result to a definition or prior self-result by construction. The arithmetic investigations rest on the stated premise but do not exhibit self-definitional loops, fitted-input predictions, or load-bearing self-citation chains. The derivation is self-contained against external benchmarks such as the continued-fraction analogy and standard MZV definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard definitions of multiple zeta values and their star variants, plus analogies to continued fractions and Cantor sets; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Standard properties of multiple zeta values and their star variants hold for the infinite case.
    Invoked when extending finite MZVs to infinite representations of reals >1.
  • ad hoc to paper Continued fraction expansions and Cantor set constructions provide valid analogies for algebraic properties of the sums and products.
    Used to motivate the conjectures on algebraic points.

pith-pipeline@v0.9.0 · 5377 in / 1194 out tokens · 33147 ms · 2026-05-14T22:28:48.783156+00:00 · methodology

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Reference graph

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