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

TL;DR

This paper generalizes rational approximations for Euler's constant and related functions, providing explicit formulas and confirming conjectures on denominators, with implications for approximations of /2 b1 a3.

Contribution

It introduces explicit constructions of rational approximations for b3, bb, and related constants, extending prior work and proving conjectures on denominators.

Findings

Explicit formulas for numerators and denominators of approximations.

Rational approximations for b3 b1 a3 and related constants.

Proof of denominators conjectures by Rivoal.

Abstract

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 $\zeta(2)-\gamma^2.$