Vol. 11, No. 2, 2022

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

Vladislav K. Salikhov and Mariya G. Bashmakova

Vol. 11 (2022), No. 2, 181–188

We prove a series of new results for the irrationality measure of some values of arctan x. Some time ago we applied symmetric complex integrals to approximate values of the form arctan (1n), where n is a natural number. This method gave new estimates for the numbers arctan 1 2 and arctan 1 3 only. To deal with some other values of arctan 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 arctan (sr) for some natural s,r , s < r.

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irrationality measure
Mathematical Subject Classification
Primary: 11J82
Received: 26 November 2021
Revised: 8 June 2022
Accepted: 22 June 2022
Published: 13 August 2022
Vladislav K. Salikhov
Department of Higher Mathematics
Bryansk State Technical University
Mariya G. Bashmakova
Department of Higher Mathematics
Bryansk State Technical University