Analytic Number Theory: Hypergeometric Functions and Irrationality Measures

@inproceedings{Viola1997AnalyticNT,
  title={Analytic Number Theory: Hypergeometric Functions and Irrationality Measures},
  author={Carlo Viola},
  year={1997}
}
Arithmetic of linear forms involving odd zeta values
The story exposed in this paper starts in 1978, when R. Apery [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apery’sExpand
Approximation Measures for Logarithms of Algebraic Numbers FRANCESCO AMOROSO-CARLO
Given a number field K and a number ) lll we say that A > 0 is a K-irrationality measure of 03BE if, for any e > 0, > ( 1 + e) p h(o) for all p E K with sufficiently large Weil logarithmic heightExpand
HYPERGEOMETRY INSPIRED BY IRRATIONALITY QUESTIONS
We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying `permutation group'Expand
Multiple Legendre polynomials in diophantine approximation
We construct a class of multiple Legendre polynomials and prove that they satisfy an Apery-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers byExpand
IRRATIONALITY AND NONQUADRATICITY MEASURES FOR LOGARITHMS OF ALGEBRAIC NUMBERS
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 aExpand
A refinement of Nesterenko’s linear independence criterion with applications to zeta values
We refine (and give a new proof of) Nesterenko’s famous linear independence criterion from 1985, by making use of the fact that some coefficients of linear forms may have large common divisors. ThisExpand
NEW IRRATIONALITY RESULTS FOR DILOGARITHMS OF RATIONAL NUMBERS
A natural method to investigate diophantine properties of transcendental (or conjecturally transcendental) constants occurring in various mathematical contexts consists in the search for sequences ofExpand
Linear independence of linear forms in polylogarithms
For x ∈ C, |x | < 1, s ∈ N, let Lis(x) be the s-th polylogarithm of x . We prove that for any non-zero algebraic number α such that |α| < 1, the Q(α)-vector space spanned by 1, Li1(α), Li2(α), . . .Expand
An essay on irrationality measures of pi and other logarithms
We present a brief survey of the methods used in deducing upper estimates for irrationality measures of the logarithm values. We particularly expose the best known estimates for $\log2$ (due to E.Expand
Irrationality Measures of log 2 and π/√3
TLDR
Using a class of polynomials that generalizes Legendre polynmials, this work unify previous works of E. V. Chudnovsky about irrationality measures of log 2 and π/√3. Expand