Geodesic
On Catalan's constant
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 159-176

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A new efficient construction of Diophantine approximations to Catalan's constant is presented that is based on the direct analysis of the representation of a hypergeometric function with specially chosen half-integer parameters as a series and as a double Euler integral over the unit cube. This allows one to significantly simplify the proofs of Diophantine results available in this domain and substantially extend the capabilities of the method. The sequences of constructed rational approximations are not good enough to prove irrationality, but the results established allow one to compare the quality of various constructions. 
@article{TM_2016_292_a9,
     author = {Yu. V. Nesterenko},
     title = {On {Catalan's} constant},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {159--176},
     publisher = {mathdoc},
     volume = {292},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_292_a9/}
}
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Yu. V. Nesterenko. On Catalan's constant. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, geometry, and number theory, Tome 292 (2016), pp. 159-176. http://geodesic.mathdoc.fr/item/TM_2016_292_a9/