Does this sum is anywhere near to help, to show that the $\gamma$ the Euler's constant is near to an irrational number?
$$\sum_{n=1}^{\infty}\left(H_n-2H_{2n-1}-\frac{1}{4n}\cdot\frac{4n-3}{2n-1}+\ln(4n-2)+\gamma\right)=0$$
where $H_n$ harmonic number