On rational approximations for some values of arctan(s/r) for natural s and r, s<r

Abstract

We prove a series of new results for the irrationality measure of some values of $\text{arctan\hspace{0.17em}}x$. Some time ago we applied symmetric complex integrals to approximate values of the form $\text{arctan}(1∕n)$, where $n\in \mathbb{N}$ is a natural number. This method gave new estimates for the numbers $\text{arctan\hspace{0.17em}}\frac{1}{2}$ and $\text{arctan\hspace{0.17em}}\frac{1}{3}$ only. To deal with some other values of $\text{arctan\hspace{0.17em}}x$ we modify the main construction. In the present paper, we consider a new integral which is based on an idea of Q. Wu and does not have a property of symmetry of the integrand. Integral construction of such a type allows us to improve estimates for the irrationality measure of some values of $\text{arctan}(s∕r)$ for some natural $s,r\in \mathbb{N}$, .