# The permutation group method for the dilogarithm

@article{Rhin2005ThePG, title={The permutation group method for the dilogarithm}, author={Georges Rhin and Carlo Viola}, journal={Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze}, year={2005}, volume={4}, pages={389-437} }

We give qualitative and quantitative improvements on all the best pre- viously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the diloga- rithm.

#### 15 Citations

NEW IRRATIONALITY RESULTS FOR DILOGARITHMS OF RATIONAL NUMBERS

- Mathematics
- 2008

A natural method to investigate diophantine properties of transcendental (or conjecturally transcendental) constants occurring in various mathematical contexts consists in the search for sequences of… Expand

Irrationality of certain numbers that contain values of the di- and trilogarithm

- Mathematics
- 2006

We prove that, for z ∈ {1/2, 2/3, 3/4, 4/5}, at least one of the two numbers is irrational.

Hypergeometric approximations to polylogarithms

- Mathematics
- 2006

We present three applications of the hypergeometric method to the arithmetic study of polylogarithm values and, in particular, of zeta values. Part 1 is joint work with Khodabakhsh and Tatiana… Expand

Approximations to-, di-and tri-logarithms Wadim Zudilin

- 2004

We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apéry-like… Expand

IRRATIONALITY AND NONQUADRATICITY MEASURES FOR LOGARITHMS OF ALGEBRAIC NUMBERS

- Mathematics
- Journal of the Australian Mathematical Society
- 2012

Abstract Let 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to a… Expand

Sums of series of Rogers dilogarithm functions

- Mathematics
- 2009

Abstract
Some sums of series of Rogers dilogarithm functions are established by Abel’s functional equation.

Approximations to -, di- and tri-logarithms

- Mathematics
- 2007

We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apery-like… Expand

Recent Diophantine results on zeta values : a survey (Analytic number theory and related topics)

- Mathematics
- 2010

After the proof by R. Apéry of the irrationality of ζ(3) in 1976, a number of articles have been devoted to the study of Diophantine properties of values of the Riemann zeta function at positive… Expand

On a continued fraction expansion for Euler's constant

- Mathematics
- 2010

Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant $\gamma$ defined by a third-order homogeneous linear recurrence. In this paper, we give… Expand

Linear Forms in Polylogarithms

- Mathematics
- ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
- 2021

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x <1$. Consider $\Phi_s(x,z) =\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch function with… Expand

#### References

SHOWING 1-4 OF 4 REFERENCES

Rational approximations to the dilogarithm

- Mathematics
- 1993

The irrationality proof of the values of the dilogarithmic function L 2 (z) at rational points z = 1/k for every integer k ∈ (−∞, −5] ∪ [7, ∞) is given. To show this we develop the method of… Expand

A Note on the Irrationality of ζ(2) and ζ(3)

- Mathematics
- 1979

At the “Journees Arithmetiques” held at Marseille-Luminy in June 1978, R. Apery confronted his audience with a miraculous proof for the irrationality of ζ(3) = l-3+ 2-3+ 3-3 + .... The proof was… Expand

The group structure for ζ(3)

- Mathematics
- 2001

1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. ..… Expand