Catalan's constant is irrational

Abstract: Whether the constant $$G=\sum_{k=0}<sup>\infty\frac{(-1)<sup>k}{(2k+1)<sup>2}=\frac1{1<sup>2}-\frac1{3<sup>2}+\frac1{5<sup>2}-\frac1{7<sup>2}+\cdots$$ introduced by Catalan in the nineteen century is irrational, is a long-standing open problem. In this paper we prove the irrationality of $G$ via using suitable weights.

• The paper proves the irrationality of Catalan’s constant is hashed conditional structure, factors showing that the initial $G=a/q$ in each step.

Main result and context

The paper proves that Catalan’s constant

$$G=\beta(2)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^2}$$

is irrational (2609.04176). Here $\beta(s)=L(s,\chi_{-4})$ is the Dirichlet beta function, so the result establishes irrationality for the first nontrivial even value of this $L$-function. Unlike $\beta(1)=\pi/4$, whose irrationality follows immediately from the transcendence of $\pi$, the arithmetic nature of $\beta(2)$ has resisted previous methods involving Euler sums, Padé approximants, and arithmetic holonomy.

The proof is conditional in structure: it assumes temporarily that $G=a/q\in\mathbb Q$ in lowest terms and constructs, for a large integer parameter $B$, a nonzero integer whose absolute value is eventually smaller than $1$. The contradiction comes from an explicit quadratic decay estimate. The central innovation is the introduction of weighted tails

$$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$

The additional factor $\beta(s)=L(s,\chi_{-4})$0 is not cosmetic. It changes the determinant structure sufficiently to make the relevant divisibility and archimedean estimates compatible within one fixed rational scalar.

The paper situates the result alongside the proof that $\beta(s)=L(s,\chi_{-4})$1 is irrational by Calegari, Dimitrov, and Tang (Calegari et al., 2024). That method supplies part of the conceptual background but does not establish the corresponding assertion for Catalan’s constant. The present argument instead modifies the auxiliary sequences and uses a weighted determinant construction.

Weighted tails and the rank theorem

The tails satisfy the elementary recurrence

$\beta(s)=L(s,\chi_{-4})$2

together with

$\beta(s)=L(s,\chi_{-4})$3

Iterating the recurrence expresses $\beta(s)=L(s,\chi_{-4})$4 as an alternating multiple of $\beta(s)=L(s,\chi_{-4})$5 plus a finite rational expression in $\beta(s)=L(s,\chi_{-4})$6. After division by $\beta(s)=L(s,\chi_{-4})$7, this yields a decomposition of $\beta(s)=L(s,\chi_{-4})$8 into a tail component and a polynomial component.

For positive integers $\beta(s)=L(s,\chi_{-4})$9, the paper defines

$L$0

and an $L$1 weighted residual matrix with entries

$L$2

The first structural result is that this matrix has full column rank $L$3.

The proof combines finite-difference annihilation with a rational-function obstruction. If a nonzero vector lies in the right kernel, the associated sequence can be written as

$L$4

where $L$5 is a polynomial of degree at most $L$6, while $L$7 is a nonzero polynomial of degree at most $L$8. Newton interpolation and the vanishing of high finite differences produce a polynomial $L$9 such that the defect

$\beta(1)=\pi/4$0

has many integer zeros. It also has zeros forced by divisibility properties of $\beta(1)=\pi/4$1 and one additional zero at $\beta(1)=\pi/4$2.

If $\beta(1)=\pi/4$3 were nonzero, its number of distinct zeros would exceed its degree. If $\beta(1)=\pi/4$4 were identically zero, the rational function $\beta(1)=\pi/4$5 would satisfy

$\beta(1)=\pi/4$6

After translation, this becomes a rational difference equation with right-hand side $\beta(1)=\pi/4$7. A pole argument shows that no rational function satisfies it: a pole of maximal real part would be forced to occur at one location, while a pole of minimal real part would be forced to occur at an incompatible translated location. This contradiction establishes full column rank.

Consequently, one can select a set $\beta(1)=\pi/4$8 with $\beta(1)=\pi/4$9 for which the residual minor $\pi$0, with $\pi$1, is nonzero. This selected minor becomes the arithmetic core of the later determinant.

Construction of the fixed scalar

The selected residual minor is completed to a square determinant by adjoining Newton interpolation columns. The resulting matrix contains three types of columns: polynomial monomials, the weighted-tail columns $\pi$2, and three binomial columns corresponding to the omitted residual rows.

Applying a lower-triangular finite-difference transformation converts the polynomial columns into a triangular block with determinant

$\pi$3

while the auxiliary binomial columns become signed unit vectors. Expanding along those columns gives the exact identity

$\pi$4

The scalar $\pi$5 is nonzero by construction.

Assuming $\pi$6, every weighted tail $\pi$7 is rational. The paper defines $\pi$8 as the denominator of $\pi$9. Prime-by-prime,

$\beta(2)$0

Thus $\beta(2)$1 measures precisely the positive part of the denominator contribution not cancelled by the residual determinant. The final contradiction depends on estimating this same scalar both arithmetically and archimedeanly; this same-scalar normalization is essential because separate estimates for numerator and denominator would not preserve the required cancellation of $\beta(2)$2 terms.

Pascal–Cauchy factorization

The residual determinant admits a Cauchy–Binet expansion. Each summand indexed by an $\beta(2)$3-element subset $\beta(2)$4 factors into a Pascal determinant, a diagonal tail contribution, and a Cauchy determinant.

The Pascal factor contains one Vandermonde determinant: $\beta(2)$5 while the Cauchy determinant contributes a second copy of the same Vandermonde. The exact absolute value of a typical summand therefore contains $\beta(2)$6, together with factorial terms, the fixed Vandermonde $\beta(2)$7, the integer polynomial factor $\beta(2)$8, and the weighted-tail factors.

The duplication of the Vandermonde is quantitatively decisive. At an odd prime power $\beta(2)$9, its valuation is represented by residue-class collisions: $G=a/q\in\mathbb Q$0 where $G=a/q\in\mathbb Q$1 counts elements of $G=a/q\in\mathbb Q$2 in the residue class $G=a/q\in\mathbb Q$3 modulo $G=a/q\in\mathbb Q$4. This converts the determinant estimate into an optimization problem over residue occupancies.

Prime-power layers and the positive-part bridge

For each odd prime power $G=a/q\in\mathbb Q$5, the paper defines a local layer $G=a/q\in\mathbb Q$6 incorporating row factorials, Vandermonde collisions, Cauchy denominators, the factors $G=a/q\in\mathbb Q$7, and the lower bounds supplied by the tail estimate. Let

$G=a/q\in\mathbb Q$8

The determinant valuation satisfies

$G=a/q\in\mathbb Q$9

The denominator layer is

$B$0

where $B$1 is the collision-counting function

$B$2

The principal local comparison is the saturation theorem: $B$3 for every odd prime power $B$4 and $B$5.

This inequality is established by comparing the minimizing configuration with the consecutive test set $B$6. The proof reduces the comparison to a combinatorial cross-collision inequality between two residue multisets. The exact formula for $B$7 gives sufficiently sharp control of the floor-function errors. Under the parameter restriction $B$8, the final lower bound is

$B$9

The restriction $1$0 therefore enters as a quantitative stability condition, not merely as a convenient normalization.

The saturation theorem yields the positive-part bridge

$1$1

The prime $1$2 contributes no positive-part height. Specifically, $1$3 is odd, while the relevant entries remain $1$4-integral under the rationality assumption, so

$1$5

This correction is important: the low-prime constant in the final estimate must be the odd-prime constant, rather than a full local constant that includes an extraneous $1$6 contribution.

A further stability lemma replaces the selected-row model by a consecutive ideal model. The two row configurations differ by only finitely many indices, and the resulting local discrepancy is bounded by

$1$7

After summation over prime powers, the total effect is $1$8. The same estimate controls the real-place normalization. This permits the asymptotic analysis to use a simpler model without changing the leading quadratic coefficient.

Small prime powers

For odd prime powers $1$9, the local optimization has a periodic structure. Writing $$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$0 and $$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$1, the residue occupancies decompose into complete blocks and a fractional terminal block. The local marginal costs form ladders whose offsets are encoded by a piecewise polynomial function $$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$2.

A periodicity identity for the integrated ladder cost gives

$$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$3

After combining the ladder contribution with the row-factorial term, the local density takes the form

$$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$4

where $$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$5 and $$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$6 is piecewise quadratic.

At the selected value $$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$7, the renormalized small-prime contribution is evaluated by reducing it to an integral involving Hurwitz zeta functions. The paper gives the certified interval

$$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$8

and therefore

$$T_m=\sum_{r=0}^{\infty}\frac{(-1)^r}{(2m+2r+1)^2}, \qquad u_m=\frac{T_m}{2m+1}.$$9

The computation is not presented as a floating-point heuristic. The integrand is partitioned into finitely many cells, with 238 raw breakpoints and 178 merged cells. On each cell, the relevant function is polynomial, and the special-function antiderivatives are evaluated using recurrence relations, Euler–Maclaurin expansions, and explicit remainder bounds. The correctness of the global proof therefore depends on the validity of this interval certificate and its exact bookkeeping.

Middle and large prime ranges

For $\beta(s)=L(s,\chi_{-4})$00, only the first $\beta(s)=L(s,\chi_{-4})$01-adic layer contributes asymptotically. The local optimization is expressed through marginal costs within residue classes. After scaling $\beta(s)=L(s,\chi_{-4})$02, the floor functions and marginal thresholds become piecewise affine. Exact subdivision of the parameter range produces 235 affine cells, on each of which the local density is affine.

At $\beta(s)=L(s,\chi_{-4})$03, the resulting integral is the rational number

$\beta(s)=L(s,\chi_{-4})$04

with numerical value

$\beta(s)=L(s,\chi_{-4})$05

The corresponding certified interval is

$\beta(s)=L(s,\chi_{-4})$06

For the large range, $\beta(s)=L(s,\chi_{-4})$07, the local density simplifies to an explicit piecewise-linear function. Its integral is

$\beta(s)=L(s,\chi_{-4})$08

Relative to the raw Cauchy–tail baseline

$\beta(s)=L(s,\chi_{-4})$09

the improvement is

$\beta(s)=L(s,\chi_{-4})$10

At $\beta(s)=L(s,\chi_{-4})$11,

$\beta(s)=L(s,\chi_{-4})$12

Prime powers $\beta(s)=L(s,\chi_{-4})$13 with $\beta(s)=L(s,\chi_{-4})$14 contribute only $\beta(s)=L(s,\chi_{-4})$15 because their number below the relevant scale is $\beta(s)=L(s,\chi_{-4})$16 and each local layer is $\beta(s)=L(s,\chi_{-4})$17. Hence the middle and large proportional ranges may be evaluated using primes, with prime-power corrections absorbed into the lower-order term.

The quadratic contradiction

The determinant and height estimates are combined using weighted prime-power summation and the prime number theorem. The leading $\beta(s)=L(s,\chi_{-4})$18 contributions cancel because the denominator layer and the archimedean normalization arise from the same fixed scalar $\beta(s)=L(s,\chi_{-4})$19.

The remaining raw quadratic coefficient is

$\beta(s)=L(s,\chi_{-4})$20

For $\beta(s)=L(s,\chi_{-4})$21, this is

$\beta(s)=L(s,\chi_{-4})$22

The local contributions produce the estimate

$\beta(s)=L(s,\chi_{-4})$23

Substitution of the certified constants gives

$\beta(s)=L(s,\chi_{-4})$24

and hence there exists $\beta(s)=L(s,\chi_{-4})$25 such that

$\beta(s)=L(s,\chi_{-4})$26

Under the rationality assumption $\beta(s)=L(s,\chi_{-4})$27, the scalar

$\beta(s)=L(s,\chi_{-4})$28

is a nonzero integer. Since $\beta(s)=L(s,\chi_{-4})$29, the factor $\beta(s)=L(s,\chi_{-4})$30 contributes only $\beta(s)=L(s,\chi_{-4})$31 to its logarithm, whereas the preceding estimate contributes $\beta(s)=L(s,\chi_{-4})$32. Consequently,

$\beta(s)=L(s,\chi_{-4})$33

For sufficiently large $\beta(s)=L(s,\chi_{-4})$34, this implies $\beta(s)=L(s,\chi_{-4})$35, contradicting the integrality of $\beta(s)=L(s,\chi_{-4})$36. Therefore $\beta(s)=L(s,\chi_{-4})$37.

Limitations and open questions

The proof is highly sensitive to the exact local optimization and numerical certification. The decisive margin is approximately $\beta(s)=L(s,\chi_{-4})$38 in the normalized quadratic coefficient, so errors in the small-prime constant, the middle-prime integral, the large-prime correction, or the same-scalar normalization would directly affect the contradiction. The paper supplies explicit interval bounds and asymptotic error estimates, but the argument still requires independent verification of a substantial finite symbolic computation involving hundreds of cells and several special-function remainder estimates.

The construction also uses the particular ratio $\beta(s)=L(s,\chi_{-4})$39. The paper does not establish whether a broader range of ratios yields a positive margin, nor whether the weighted-tail method can produce stronger irrationality measures or linear independence results involving $\beta(s)=L(s,\chi_{-4})$40, $\beta(s)=L(s,\chi_{-4})$41, and related $\beta(s)=L(s,\chi_{-4})$42-values. More specifically, it leaves open whether analogous weight choices can treat other even Dirichlet $\beta(s)=L(s,\chi_{-4})$43-values for which the corresponding unweighted tail determinants do not provide sufficient local divisibility.

Conclusion

The paper proves the irrationality of Catalan’s constant by combining weighted alternating tails, a full-rank finite-difference residual matrix, an exact Pascal–Cauchy determinant factorization, prime-power valuation estimates, and a same-scalar archimedean contradiction. The key quantitative fact is the strict negative quadratic coefficient

$\beta(s)=L(s,\chi_{-4})$44

which dominates the $\beta(s)=L(s,\chi_{-4})$45 contribution arising from a hypothetical denominator $\beta(s)=L(s,\chi_{-4})$46. The method’s main technical contribution is not merely the determinant construction, but the compatibility of its local denominator layers with the same fixed scalar used in the real-place estimate.