Catalan’s constant: a determinant approach to irrationality

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Family 005 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Consider the alternating sum

$$G=1-\frac1{3^2}+\frac1{5^2}-\frac1{7^2}+\cdots =\sum_{j=0}^{\infty}\frac{(-1)^j}{(2j+1)^2}.$$

This is Catalan’s constant, approximately $0.916$. Its definition is elementary, and it can be calculated to very high precision. Yet computing digits does not answer a basic arithmetic question: is it a ratio of two integers?

Family 005 consists of one manuscript claiming that Catalan’s constant is irrational. Its strategy compares two kinds of size estimates for a sequence of determinants. If $G$ were rational, prime-factor information would prevent these determinants from becoming too small. An integral representation is then used to show that they become smaller than that arithmetic bound permits.

The difficult part is arranging both estimates for the same nonzero quantities. This is a useful example of a general lesson in irrationality proofs: accurate approximation alone is not enough; one must also control the arithmetic cost of that accuracy.

2. Background and essential definitions

A real number is rational if it equals $a/b$ for integers $a,b$ with $b\ne0$. Otherwise it is irrational. Transcendence is stronger: a transcendental number is not a root of any nonzero integer-coefficient polynomial. The manuscript claims irrationality, not transcendence.

Catalan’s constant is the value at two of the Dirichlet beta function,

$$\beta(s)=\sum_{j=0}^{\infty}\frac{(-1)^j}{(2j+1)^s}.$$

At positive odd integers, beta values are known rational multiples of powers of $\pi$, for example $\beta(1)=\pi/4$ and $\beta(3)=\pi^3/32$. Even arguments present a different arithmetic problem. The alternating sign pattern comes from the character modulo four that is one on integers congruent to one, minus one on those congruent to three, and zero on even integers.

One classical irrationality criterion seeks integers $A_m,B_m$ such that

$$0<|A_mG-B_m|\longrightarrow0.$$

If $G=a/b$, then $b(A_mG-B_m)$ would be a nonzero integer tending to zero, which is impossible. The nonzero condition is essential.

This criterion is stronger than $B_m/A_m\to G$. For instance, an approximation error of order $A_m^{-0.6}$ tends to zero, but multiplication by $A_m$ gives a quantity of order $A_m^{0.4}$, which need not be small. Fast numerical convergence and useful integer linear forms are different achievements.

A determinant combines entries of a square matrix into a number that vanishes exactly when its rows are linearly dependent. The manuscript replaces individual linear forms by determinants of growing size. A moment here means an integral of a polynomial against a fixed kernel; assembling many moments creates a matrix whose determinant has both an arithmetic description and an analytic one.

3. Development history and previous partial results

Apéry’s proof of the irrationality of $\zeta(3)$, followed by Beukers’s integral proofs for $\zeta(2)$ and $\zeta(3)$, demonstrated how recurrence relations and multiple integrals can yield small integer linear forms. Their success encouraged analogous constructions for other special values.

For beta values, Rivoal and Zudilin proved that infinitely many even values are irrational and that at least one among $\beta(2),\beta(4),\ldots,\beta(14)$ is irrational. Later work reduced the finite list, ultimately to the five values through $\beta(10)$, as recorded by Lai and Zhou. Such statements do not identify which member of the list is irrational.

For Catalan’s constant specifically, Zudilin constructed an Apéry-like recurrence and continued fraction. The rational approximations converge rapidly, but the resulting integer linear forms fail to decay after denominators are cleared. Rivoal and then Krattenthaler–Rivoal established important cancellations in those denominators. They sharpened the arithmetic estimates without overcoming the entire obstruction.

A later determinantal approach of Zudilin compared denominator growth with integral estimates on a quadratic scale in matrix size. For positive moment matrices, positivity provides nonvanishing. The Catalan construction considered there still suffered from excessive denominator growth. The present manuscript uses a different, signed mixture of moments, requiring a separate nonvanishing argument.

Nearby successes also need careful separation. Calegari, Dimitrov, and Tang established linear independence involving a character of conductor three, whereas Catalan’s constant uses conductor four. Irrationality of a 2-adic analogue likewise concerns a different number system and does not settle the real constant. Finally, M1 notes an earlier 2026 preprint by Zhi-Wei Sun announcing the same qualitative conclusion; it states that it does not use that preprint. The digest therefore does not assign priority to this manuscript.

4. What the new manuscripts claim

M1, Theorem 1.1, states simply

$$G\notin\mathbb Q.$$

There are no conditional hypotheses in that theorem. The manuscript describes a qualitative irrationality argument, without claiming a bound on how well $G$ can be approximated by arbitrary rational numbers.

Its central objects are determinants $\Delta_N$ of size $48N$, for positive integers $N$. Define the normalized logarithm

$$L_N=\frac{\log|\Delta_N|}{(48N)^2}-\frac12\log2.$$

Under the temporary hypothesis that $G$ is rational, M1 claims two arithmetic facts: the determinants are rational, and $\Delta_p\ne0$ for every sufficiently large prime $p$. On any sequence of nonzero determinants its denominator estimates give

$$\liminf_{N\to\infty}L_N>-2.29084.$$

An independent real-variable estimate gives

$$\limsup_{N\to\infty}L_N \le-2.290939875.$$

Applying both bounds along the prime sequence produces the stated contradiction. The numerical separation is small but strict: the proposed upper bound is below the proposed lower bound.

M1 also draws a geometric corollary. With sectional curvature normalized to $-1$, Agol’s theorem identifies $4G$ as the minimum volume of an orientable, complete, finite-volume hyperbolic three-manifold with exactly two cusps. A cusp is an end extending to infinity while contributing finite volume. Irrationality of $G$ therefore makes this minimum irrational. The manuscript additionally obtains irrationality for arithmetic hyperbolic three-orbifold volumes defined over $\mathbb Q(i)$, because their known volume formulas are positive rational multiples of $G$.

5. Why the advance matters

Provisional importance: 4/5. Catalan’s constant is a prominent elementary special value whose arithmetic has resisted methods successful for related constants. Identifying this individual number as irrational would resolve a longstanding question rather than merely guarantee an unspecified irrational member of a family.

The proposed method is also significant. It combines carefully engineered cancellation of unwanted constants, prime-by-prime denominator bounds, algebraic nonvanishing, and real analytic estimates. A successful construction of this kind could inform attempts on other special values, although no general extension is established here.

The geometric consequences illustrate how a question about a short alternating series can affect the arithmetic nature of volumes. The rating reflects this combination of centrality within irrationality theory and broader connections, conditional on the stated result.

6. The main idea

First remove the unwanted constant

The manuscript starts from two elementary integration kernels. Their moments are rational linear combinations of three quantities:

$$1,\qquad G,\qquad \zeta(2)=\frac{\pi^2}{6}.$$

Assuming $G$ rational would not make such moments rational if a $\zeta(2)$ term remained. The polynomial rows are therefore designed so that two expressions have many equal initial Taylor coefficients. This high-order agreement makes the unwanted $\zeta(2)$ coefficients cancel in the mixed entries.

The rows are built from Chebyshev polynomials, characterized by $T_d(\cos\theta)=\cos(d\theta)$. Their integral coefficients and recurrence relations give simultaneous control over Taylor behavior and arithmetic divisibility. They serve more than an approximation role: their arithmetic structure is used later in the denominator estimates.

After cancellation, each matrix entry lies in $\mathbb Q+\mathbb QG$. Thus the hypothesis $G\in\mathbb Q$ makes the whole determinant rational. It need not be a linear form in $G$; rationality of its entries is what the argument needs.

Denominators set a lower limit on size

If a nonzero rational number has denominator dividing a positive integer $D$, its absolute value is at least $1/D$. More generally, for nonzero rational $r$, write $v_p(r)$ for the exponent of $p$ in its prime factorization, allowing negative exponents. Then

$$\log|r|=\sum_p v_p(r)\log p.$$

Only finitely many terms are nonzero. This relation lets the manuscript convert information at finite primes into an ordinary real lower bound for $|\Delta_N|$.

At the large odd primes contributing to the leading estimate, the construction has denominator layers involving both $p^2$ and $p$. M1 tracks both; controlling only the first would not yield its final inequality. The prime two receives a separate estimate. The only prime-distribution input stated for this stage is the ordinary prime number theorem, not the new zero-free claims in family 003.

Make sure the determinant does not vanish

Because the moments are mixed with signs, positivity cannot simply certify a nonzero determinant. M1 instead specializes the scale to a large prime $N=p$.

Modulo $p$, taking a $p$-th power simplifies polynomial expressions through the identity $(x+y)^p=x^p+y^p$. Together with changes of polynomial bases, this reduces the growing nonvanishing problem to three fixed rational matrices. The manuscript supplies exact finite certificates for those matrices and uses them to claim nonvanishing for all sufficiently large prime scales. This is an arithmetic nonvanishing argument, not a claim that every determinant is positive.

The integral description makes it too small

An integration identity rewrites the determinant as a multiple integral involving products of differences between nodes. Such products are Vandermonde factors, familiar from polynomial interpolation. Their logarithms are sums of terms such as $\log|x_i-x_j|$, which can be treated as interactions among points.

The construction’s rational parametrization has two branches. M1 retains both, using bounded holomorphic interpolation in one range of configurations and a determinant norm bound in the other. The interpolation estimate is formulated to remain uniform when nodes approach each other; dividing by a small Vandermonde determinant would otherwise lose control.

The remaining optimization is expressed through Chebyshev expansions of logarithmic kernels and explicit rational trial data. M1 reports certificates enclosing stationary points and controlling complete intervals around them. These data supply the stated strict upper bound, rather than merely plotting a likely maximum. The digest describes that role without independently checking the certificates.

All three ingredients are needed together: rationality under the hypothesis, nonvanishing on an unbounded sequence, and incompatible size estimates on that same sequence.

7. Scope, limitations, and open questions

The conclusion is qualitative irrationality. It does not establish transcendence of $G$, algebraic independence from $\pi$, or an irrationality measure controlling every sufficiently accurate rational approximation. The manuscript explicitly says that the latter would require additional information.

The prime-scale nonvanishing assertion is made under the rationality hypothesis used for contradiction. One should not silently replace it by unconditional positivity for all $N$. Likewise, the precision of the displayed constants is part of the source’s claimed estimates; their appearance here is not an independent numerical verification.

Questions remain about the other individual even beta values, the extent to which similar determinant constructions can isolate them, and whether a quantitative approximation theorem can accompany this qualitative result. The hyperbolic-volume corollary concerns the stated geometric classes and normalization, not arbitrary hyperbolic volumes.

8. Sources and further reading

M1’s introduction states the approximation obstruction and determinant strategy. Sections 2–4 address cancellation, denominator control, and nonvanishing; Sections 5–7 develop the analytic bound; Section 8 gives the final comparison and geometric consequences.

The historical papers in Section 3 provide the successive approximation and denominator results. Agol’s two-cusped volume theorem supplies the geometric identification used in the corollary. For optional background, see Catalan’s constant and Chebyshev polynomials.

Source manuscripts

• M1. OpenAI, Catalan’s constant is irrational (September 24, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.