Rational Approximations to Values of the Digamma Function and a Conjecture on Denominators

摘要

We explicitly construct rational approximations to the numbers ln(b) - ψ(a + 1), where ψ is the logarithmic derivative of the Euler gamma function. We prove formulas expressing the numerators and the denominators of the approximations in terms of hypergeometrie sums. This generalizes the Aptekarev construction of rational approximations for the Kuler constant γ. As a consequence, we obtain rational approximations for the numbers π/2 ± γ. The proposed construction is compared with rational Rivoal approximations for the numbers γ + In(b). We verify assumptions put forward by Rivoal on the denominators of rational approximations to the numbers γ + In(6) and on the general denominators of simultaneous approximations to the numbers γ and ζ(2) - γ~2.