New 3 and 6-Term Functional Dilogarithm Equations from Beta-Type Integrals

Cetin Hakimoglu-Brown

arxiv: 2604.24588 · v3 · pith:AHPREZZQnew · submitted 2026-04-27 · 🧮 math.CA

New 3 and 6-Term Functional Dilogarithm Equations from Beta-Type Integrals

Cetin Hakimoglu-Brown This is my paper

Pith reviewed 2026-05-07 17:44 UTC · model grok-4.3

classification 🧮 math.CA

keywords dilogarithm identitiesfunctional equationsarctangent integralsLoxton-Lewin identityBytsko conjecturesdilogarithm laddersBloch-Wigner function

0 comments

The pith

The ratio of a sextic arctangent integral to a cubic one equals a rational constant and generates new dilogarithm functional equations.

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

The paper shows that dividing a certain sextic arctangent integral by a cubic one produces exactly a rational number. This single equality is used to build fresh 3-term and 6-term functional equations satisfied by the dilogarithm. Those equations in turn deliver an analytic proof for a Loxton-Lewin identity, establish two previously unknown quartic-base dilogarithm ladders, and settle several 2-term identities that Bytsko had conjectured. The work also extends an earlier result on the Bloch-Wigner dilogarithm. A reader would care because these identities connect special values of polylogarithms to algebraic numbers and appear in many areas of number theory.

Core claim

Showing that the ratio of the sextic and cubic arctangent integrals equals a rational constant allows the construction of new 3- and 6-term functional equations. From these the authors obtain an analytic proof of the Loxton-Lewin identity, derive a pair of quartic-base dilogarithm ladders, prove the conjectured 2-term dilogarithm identities of Bytsko, and extend his result for the Bloch-Wigner function.

What carries the argument

The ratio of the specific sextic arctangent integral to the cubic arctangent integral, evaluated as a rational constant that generates the functional equations.

Load-bearing premise

That the ratio of the given sextic arctangent integral to the cubic one is precisely the stated rational constant.

What would settle it

High-precision numerical computation of both the sextic and cubic integrals to check whether their ratio exactly matches the claimed rational value.

read the original abstract

Building on results by Abouzahra and Lewin, McIntosh, and Kirilov we derive new functional dilogarithm equations and consequent diologarithim ladders. By showing that the ratio of a pair of sextic and cubic integrals equals a rational constant, we construct new 3- and 6-term functional equations, from which we derive an analytic proof of an identity by Loxton-Lewin, as well as a pair of quartic-base dilogarithm ladders, also believed to be new, building on Loxton's result. Finally, we prove conjectured 2-term dilogarithm identities of Bytsko, and extend his result for the Bloch-Wigner function using the above methods.

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 derives new dilogarithm equations from an integral ratio and proves some conjectures, but the soundness depends on verifying the central integral evaluation.

read the letter

The paper's main contribution is deriving new 3-term and 6-term dilogarithm functional equations, along with quartic ladders, from the claim that the ratio of a sextic arctangent integral to a cubic one is a fixed rational constant. It then uses these to give analytic proofs of a Loxton-Lewin identity and some 2-term identities conjectured by Bytsko, plus an extension involving the Bloch-Wigner function. What stands out is the systematic way it turns the integral ratio into functional equations. This seems to produce results that go beyond the cited work by Abouzahra-Lewin, McIntosh, Kirilov, and Loxton. Providing analytic proofs instead of leaving things as conjectures is a clear step forward in this area. The soft spot is the foundation: that integral ratio. The paper treats it as exactly rational, but one would need to see the explicit evaluation to confirm there are no problems with branch cuts or convergence. If that step holds, the rest follows reasonably. The circularity is low because it starts from integrals rather than assuming the identities. This kind of work is for researchers who care about explicit dilogarithm relations, perhaps for use in evaluating special values or in physical applications. A reader who knows the standard ladders and wants to see new ones or verified proofs would get value from it. It is solid enough to deserve a serious referee, who could check the integral details and the novelty of the equations. I recommend putting it through peer review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The paper claims to derive new 3- and 6-term dilogarithm functional equations by establishing that the ratio of a specific sextic arctangent integral to a cubic arctangent integral equals a rational constant. Building on prior results of Abouzahra-Lewin, McIntosh, and Kirilov, these equations are used to obtain an analytic proof of the Loxton-Lewin identity, construct new quartic-base dilogarithm ladders, prove Bytsko's conjectured 2-term identities, and extend the result to the Bloch-Wigner function.

Significance. If the central integral-ratio evaluation is correct and free of hidden analytic assumptions, the work supplies a constructive integral-based route to dilogarithm identities that were previously conjectural or only numerically verified. The analytic proof of the Loxton-Lewin relation and the new ladders constitute concrete additions to the literature on polylogarithm functional equations.

major comments (2)

[Section deriving the integral ratio (preceding the construction of the 3-term equation)] The evaluation that the ratio of the sextic arctangent integral to the cubic arctangent integral equals the stated rational constant is the sole load-bearing step for every subsequent identity. The manuscript must supply a fully self-contained derivation of this ratio (including explicit treatment of improper-integral convergence, branch choices for arctan, and any interchange of limits or integrals) rather than relying on citations alone; without it the derived 3- and 6-term equations lose their foundation.
[Section containing the analytic proof of the Loxton-Lewin identity] In the derivation of the Loxton-Lewin identity from the new 3-term functional equation, the parameter choices that reduce the general equation to the specific Loxton-Lewin form must be stated explicitly, together with verification that no extraneous terms survive after substitution.

minor comments (2)

[Introduction and the integral-ratio section] Clarify the precise rational value asserted for the integral ratio at its first appearance and restate it when it is used to construct each functional equation.
[Section presenting the new functional equations] Add a short table or explicit list of the new 3- and 6-term equations with their arguments to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comments. We address each point below and will revise the paper accordingly to improve self-containment and explicitness of the derivations.

read point-by-point responses

Referee: [Section deriving the integral ratio (preceding the construction of the 3-term equation)] The evaluation that the ratio of the sextic arctangent integral to the cubic arctangent integral equals the stated rational constant is the sole load-bearing step for every subsequent identity. The manuscript must supply a fully self-contained derivation of this ratio (including explicit treatment of improper-integral convergence, branch choices for arctan, and any interchange of limits or integrals) rather than relying on citations alone; without it the derived 3- and 6-term equations lose their foundation.

Authors: We agree that the integral-ratio evaluation is foundational and that the present version relies on the cited works of Abouzahra-Lewin, McIntosh, and Kirilov. In the revised manuscript we will insert an expanded, self-contained derivation of the ratio (either in the main text or as a dedicated appendix). This will include: (i) explicit verification of convergence for the improper integrals at the endpoints, (ii) a clear statement of the principal branch chosen for each arctangent, and (iii) a justification, with dominated-convergence or uniform-convergence arguments, for all interchanges of limits and integrals. The new material will build directly on the cited references but will not require the reader to consult them for the core steps. revision: yes
Referee: [Section containing the analytic proof of the Loxton-Lewin identity] In the derivation of the Loxton-Lewin identity from the new 3-term functional equation, the parameter choices that reduce the general equation to the specific Loxton-Lewin form must be stated explicitly, together with verification that no extraneous terms survive after substitution.

Authors: We accept that the reduction step would benefit from greater explicitness. In the revised version we will list the precise numerical values assigned to each free parameter in the general 3-term equation, display the resulting simplified expression, and then verify term-by-term that every non-Loxton-Lewin term cancels or vanishes identically. This will be presented immediately before the final statement of the identity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on independent integral evaluation.

full rationale

The paper's central step is the explicit evaluation of the ratio of one sextic arctangent integral to one cubic arctangent integral equaling a stated rational constant, from which 3- and 6-term functional equations are constructed. This ratio is obtained by direct computation building on external prior results (Abouzahra-Lewin, McIntosh, Kirilov) rather than by fitting parameters to the target dilogarithm identities or by self-definition. Subsequent derivations of the Loxton-Lewin identity, quartic ladders, and Bytsko 2-term identities follow algebraically from those equations without reducing back to the input integrals by construction. No self-citations appear as load-bearing premises, and the cited results are independent of the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard properties of the dilogarithm and on the evaluation of specific arctangent integrals. No free parameters, ad-hoc axioms, or new invented entities are mentioned in the abstract.

axioms (1)

standard math Standard analytic continuation and branch-cut properties of the dilogarithm function hold for the arguments appearing in the new identities.
Invoked implicitly when moving from integral representations to functional equations.

pith-pipeline@v0.9.0 · 5413 in / 1381 out tokens · 171586 ms · 2026-05-07T17:44:59.201714+00:00 · methodology

New 3 and 6-Term Functional Dilogarithm Equations from Beta-Type Integrals

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)