Rational approximations for values of the digamma function and a denominators conjecture

Details Rational approximations for values of the digamma function and a denominators conjecture Rational approximations for values of the digamma function and a denominators conjecture

2010-04-05

arxiv.org/abs/1004.0578v1

Mathematics

Number Theory

19 pages, in Russian

Scientific paper

In 2007, A.I.Aptekarev and his collaborators discovered a sequence of rational approximations to Euler's constant $\gamma$ defined by a linear recurrence. In this paper, we generalize this result and present an explicit construction of rational approximations for the numbers $\ln(b)-\psi(a+1),$ $a, b\in {\mathbb Q},$ $b>0, a>-1,$ where $\psi$ defines the logarithmic derivative of the Euler gamma function. We prove exact formulas for denominators and numerators of the approximations in terms of hypergeometric sums. As a consequence, we get rational approximations for the numbers $\pi/2\pm\gamma.$ We compare the results obtained with those of T. Rivoal for the numbers $\gamma+\ln(b)$ and prove denominators conjectures proposed by Rivoal for denominators of rational approximations for $\gamma+\ln(b)$ and common denominators of simultaneous approximations for the numbers $\gamma$ and $\zeta(2)-\gamma^2.$

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