Geodesic
Quadratic irrationality exponents of certain numbers
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2013), pp. 25-29

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The paper presents upper estimates for the non-quadraticity measure of the numbers $\sqrt{2k+1}\ln\bigr((k+1-\sqrt{2k+1})/k\bigl)$ and $\sqrt{2k-1}\operatorname{arctg}\bigr(\sqrt{2k-1}/(k-1)\bigl)$, where $k\in\mathbb{N}$. In particular, we improved an upper estimate for the non-quadraticity measure of $\ln2$. 
@article{VMUMM_2013_5_a3,
     author = {A. A. Polyanskii},
     title = {Quadratic irrationality exponents of certain numbers},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {25--29},
     publisher = {mathdoc},
     number = {5},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2013_5_a3/}
}
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A. A. Polyanskii. Quadratic irrationality exponents of certain numbers. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2013), pp. 25-29. http://geodesic.mathdoc.fr/item/VMUMM_2013_5_a3/