On irrationality of Euler's constant and related asymptotic formulas

Abstract

By defining $$I_n:=\int_{0}^{1}\int_{0}^{1} \frac{(x(1-x)y(1-y))^n}{(1-xy)(-\log xy)}\ dx dy$$ Sondow (see [2]) proved that $$I_n=\binom{2n}{n} \gamma+L_n-A_n$$ We prove asymptotic formula for $L_n$ and $A_n$ as $n\to\infty$, $$ L_n=\binom{2n}{n}\left(\log \left( {\frac{{3n}}{2}} \right) +\mathcal{O}\!\left( {\frac{1}{n}} \right)\right)$$ and $$A_n\sim\frac{4^n}{\sqrt{\pi n}}\left(\gamma+\ln\frac32+\ln n\right)$$ Using the sufficient condition for irrationality criteria of Euler's constant due to Sondow, we prove that $\gamma$ is irrational.