Improved rational approximations to Catalan's constant

Abstract

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