Improved rational approximations to Catalan's constant

Payman Eskandari

paper

Improved rational approximations to Catalan's constant

arXiv:2609.26354

Abstract

The works of Rivoal, Zudilin, and Krattenthaler and Nesterenko give an explicit sequence $p_{n} / q_{n}$ ( $p_{n}, q_{n} \in Z, q_{n} > 0$ ) of rational approximations to Catalan's constant $G = 1 - 1/ 3^{2} + 1/ 5^{2} - 1/ 7^{2} + 1/ 9^{2} - \dots$ satisfying $∣ G - p_{n} / q_{n} ∣ \leq 1/ q_{n}^{0.52}$ for sufficiently large $n$ . We build on their works to give an explicit sequence $p_{n} / q_{n}$ ( $p_{n}, q_{n} \in Z, q_{n} > 0$ ) with $∣ G - p_{n} / q_{n} ∣ \leq 1/ q_{n}^{0.62}$ for sufficiently large $n$ .

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