arxiv: 2605.08899 · v1 · submitted 2026-05-09 · 🧮 math.NT

Recognition: no theorem link

Multiple integral representations of the Catalan's constant

Emilio G\'omez-D\'eniz, Jos\'e Mar\'ia Sarabia

Pith reviewed 2026-05-12 01:11 UTC · model grok-4.3

classification 🧮 math.NT

keywords Catalan's constantintegral representationsLerch functionmultiple integralsdouble integralspecial functions

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The pith

Catalan's constant admits new single-integral and higher-dimensional multiple-integral representations derived from a general theorem.

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

The paper derives several novel integral representations for Catalan's constant. It starts with a double integral, then proves a general theorem that reduces the constant to a single integral. It further extends the approach to multiple integrals in two or more dimensions by means of the Lerch function, supplying examples for each case. A sympathetic reader would care because integral forms often permit new analytic manipulations or numerical approaches to this irrationality-related constant.

Core claim

We present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem that enables the representation of Catalan's constant in terms of a single integral. Finally, we provide a multiple integral representation of Catalan's constant in dimensions greater than or equal to two using the Lerch function.

What carries the argument

A general theorem obtained from an initial double-integral form that allows Catalan's constant to be written as a single integral, extended to higher dimensions via the Lerch function.

If this is right

Catalan's constant equals a concrete single integral obtained from the general theorem.
It also equals multiple integrals of dimension two or higher expressed using the Lerch function.
The representations hold with illustrative examples that confirm the equalities.
The approach begins from a double-integral identity and proceeds by successive generalization.

Where Pith is reading between the lines

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

The single-integral form could be used to obtain new series expansions or asymptotic behaviours for the constant.
Higher-dimensional versions might connect Catalan's constant to integrals appearing in other branches of analysis.
Alternative quadrature rules based on these integrals could serve as independent numerical checks.

Load-bearing premise

The defining series for Catalan's constant can be interchanged with the proposed integrals, and the Lerch function satisfies the necessary analytic continuation and convergence conditions.

What would settle it

A high-precision numerical quadrature of one of the stated single or multiple integrals that yields a value differing from the known approximation 0.915965594... by more than quadrature error.

read the original abstract

In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem that enables the representation of Catalan's constant in terms of a single integral. Finally, we provide a multiple integral representation of Catalan's constant in dimensions greater than or equal to two using the Lerch function. The results are accompanied by illustrative examples.

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 adds a handful of integral representations for Catalan's constant via the Lerch function, but the work is incremental and does not introduce new techniques or resolve open questions.

read the letter

The main point is that the authors convert the alternating series for G into a double integral, derive a general single-integral theorem from it, and then produce higher-dimensional versions using the Lerch transcendent, with some examples attached. That is the entire contribution in outline form. They follow the usual route of inserting integral kernels and interchanging sum and integral under standard convergence conditions. The steps look logically ordered and the Lerch function is applied in its familiar domain, which is a plus. The examples help make the claims concrete. The derivations appear to rest on established properties rather than new machinery. The soft spot is that integral representations of this constant are already plentiful in the literature, so these versions do not stand out as a first-principles advance or a unification. The abstract does not claim to prove irrationality or supply closed forms, and the novelty is mainly in the specific forms obtained. A referee would still need to verify the precise justification for each interchange and any analytic continuation steps, but the stress-test outline shows no internal contradictions. This paper is aimed at readers who collect different integral expressions for classical constants, perhaps for computational experiments or further formal manipulation. It is not aimed at people looking for major number-theoretic results. The work is coherent on its own terms and the claims are modest, so it deserves a serious referee rather than a desk rejection. I would recommend sending it to review.

Referee Report

0 major / 3 minor

Summary. The paper derives novel integral representations for Catalan's constant G. It begins with a double-integral form obtained from the alternating series definition, establishes a general theorem yielding single-integral representations as a consequence, and constructs multiple-integral representations (n ≥ 2) via the Lerch transcendent, accompanied by illustrative examples.

Significance. If the claimed identities hold, the systematic progression from double to single to higher-dimensional integrals supplies new analytic tools for G that may aid numerical evaluation or connections to other special functions. The explicit use of the Lerch transcendent in the multiple-integral case is a concrete extension of known techniques.

minor comments (3)

[General theorem] The general theorem (following the double-integral result) should state the precise conditions on the parameters under which the interchange of sum and integral is justified, including any appeal to dominated convergence or Fubini-Tonelli.
[Multiple integral representations] In the multiple-integral section, the domain of absolute convergence for the Lerch function should be explicitly delimited before invoking meromorphic continuation, with a brief remark on why the resulting expression remains valid for the specific series of G.
[Examples] The illustrative examples would benefit from a short numerical check (e.g., quadrature versus known value of G to 10 decimals) to illustrate practical utility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The referee's summary accurately captures the structure of the paper, including the derivation of the double-integral representation from the alternating series, the general theorem for single-integral forms, and the extension to multiple integrals via the Lerch transcendent with examples.

Circularity Check

0 steps flagged

No significant circularity; derivations start from the defining series

full rationale

The paper begins with the standard alternating series for Catalan's constant G = ∑ (-1)^k/(2k+1)^2, inserts integral kernels, interchanges sum and integral under justified convergence conditions, and obtains representations via the Lerch transcendent in its standard domain. The double-integral result leads to a single-integral theorem and higher-dimensional forms as direct consequences, without any parameter fitting, self-referential definitions, or load-bearing self-citations that reduce the claimed identities to their inputs. All steps are explicit derivations from the series definition plus analytic continuation, making the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the ability to interchange summation and integration, the known integral representation of the Lerch transcendent, and convergence of the resulting expressions for the specific parameter values that recover Catalan's constant. No free parameters are introduced. No new entities are postulated.

axioms (2)

standard math Interchange of sum and integral is valid under the stated conditions
Invoked when converting the defining series of G into the proposed double integral
domain assumption Lerch transcendent satisfies the required functional equations and analytic continuation
Used to extend the representation to multiple integrals in dimensions >=2

pith-pipeline@v0.9.0 · 5364 in / 1337 out tokens · 38685 ms · 2026-05-12T01:11:21.758149+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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[3]

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[7]

Catalan’s Constant

Marichev, O., Sondow, J., Weisstein, Eric W. Catalan’s Constant. FromMathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/CatalansConstant.html

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Mortini, R. (2024). Yet Another Integral Representation of Catalan’s Constant.The American Mathematical Monthly, 131(2), 130–130.https://doi.org/10.1080/00029890. 2023.2274242

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[9]

Safonova, T. A. (2025). On old and new Formulas for the Catalan and Apéry constants.Mathematical Notes, 117(3), 484–488.https://doi.org/10.1134/ S0001434625030149

work page 2025
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Stewart, S. M. (2020). A Catalan constant inspired integral odyssey.The Mathematical Gazette, 104(561), 449–459.https://doi.org/10.1017/mag.2020.99 Appendix: expressions of the Lerch function in Theorem 3 In this Appendix we include some expressions of the Lerch function in Theorem 3, for values Φ(−z, j, 1 2)withj=−1, . . . ,−8, which correspond to multip...

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