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Catalan's Irrationality

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This paper proves the irrationality of Catalan’s constant using advanced mathematical techniques and determinant constructions.

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=β(2)=∑k=0∞(−1)k(2k+1)2G=\beta(2)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^2} is irrational (2609.04176). Here β(s)=L(s,χ−4)\beta(s)=L(s,\chi_{-4}) is the Dirichlet beta function, so the result establishes irrationality for the first nontrivial even value of this LL-function. Unlike β(1)=π/4\beta(1)=\pi/4, whose irrationality follows immediately from the transcendence of π\pi, the…

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