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Conference in memory of A. A. Karatsuba on number theory and applications, 2015
January 30, 2015 10:35–11:00, Moscow, Steklov Mathematical Institute of the Russian Academy of Sciences

On Catalan's constant

Yu. V. Nesterenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
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Yu. V. Nesterenko

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Abstract: Catalan's constant $G=0.915965594177\ldots$, which is one of the classical mathematical constants, can be defined by each of the following series:
$$ G=\sum\limits_{k = 0}^{+\infty}\frac{(-1)^{k}}{(2k+1)^{2}} = \frac{1}{2}\sum\limits_{k = 0}^{+\infty} \frac{4^{k}}{(2k+1)^{2}}\binom{2k}{k}^{-1}. $$
It is supposed that this constant is irrational, but this fact is still unproved. Practically, all the proofs of the irrationality of some numbers are based on the construction of sufficiently close approximations to these numbers. During the last years, some new approaches to the construction of rational numbers approximating Catalan's constatnt are given by the works of W.W.Zudilin, T.Rivoal, K.Kratentaller.
It is known that for every real $\alpha$ the inequality
$$ |\alpha- \frac{p}{q}| \leqslant \frac{1}{q} $$
has infinitely many solutions in the rational numbers $p/q$. To prove the irrationality of $\alpha$, it is sufficient to prove the above inequality with $\varepsilon q^{-1}$ in the right-hand side for any $\varepsilon > 0$. The cases when it is possible to do it, are very rare. For expample, we are unable to prove this for the Catalan's constant.
In the talk, we introduce a new construction of diophantine approximations to $G$ which is based on the representation of the hypergeometric function $_{3}F_{2}$ with special arguments as a series and as double integral of Euler's type over a unit cube. The direct using of the function $_{3}F_{2}$ allow one to simplify essentially the proofs of some known results in this topic and to expand essentially the possibilities of this method.
Particulary, this method allow one to construct effectively an infinite sequence of rational numbers satisfying the inequality
$$ |\alpha- \frac{p}{q}| \leqslant q^{- 1/2}. $$
Of course, it is unsufficient for the proof of the irrationality of $G$. However, the results of such type allow us to compare the quality of different approximating constructions.
Some modification of the method leads us to more precise result with a bound $q^{-11/20}$ in the right-hand side. Unfortunately, the proof of last assertion uses one special property of our construction, which is checked by computer calculations for the large set of parameters. However, in general case it is still unproved.

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