Arithmetic of linear forms involving odd zeta values

@article{Zudilin2002ArithmeticOL,
  title={Arithmetic of linear forms involving odd zeta values},
  author={Wadim Zudilin},
  journal={Journal de Theorie des Nombres de Bordeaux},
  year={2002},
  volume={16},
  pages={251-291}
}
  • W. Zudilin
  • Published 18 June 2002
  • Mathematics
  • Journal de Theorie des Nombres de Bordeaux
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’s discovery we refer to the review [Po].) Although the irrationality of the even zeta values ζ(2), ζ(4), . . . for that moment was a classical result (due to L. Euler and F. Lindemann), Apery’s proof allows one to obtain a quantitative version of his result, that is, to evaluate irrationality exponents: 

One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means

Available proofs of result of the type "at least one of the odd zeta values $\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational" make use of the saddle-point method or of linear independence criteria,

Irrationality of some p-adic L-values

We give a proof of the irrationality of the p -adic zeta-values ζ p ( k ) for p = 2 , 3 and k = 2 , 3. Such results were recently obtained by F.Calegari as an application of overconvergent p -adic

Irrationality of some p-adic L-values

We give a proof of the irrationality of p-adic zeta-values ζp(k) for p = 2,3 and k = 2, 3. Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In

Irrationality proofs for zeta values, moduli spaces and dinner parties

A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality

On the irrationality measure for a q-analogue of ζ(2)

A Liouville-type estimate is proved for the irrationality measure of the quantities ζq(2) = ∞ ∑ n=1 q (1− qn)2 with q−1 ∈ Z \ {0,±1}. The proof is based on the application of a q-analogue of the

Keywords. Irrationality; Linear independence over a field; Measures of irrationality and of transcendence; Distribution modulo one.

In this paper we generalize Nesterenko's criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both

N T ] 2 0 Ju n 20 06 Irrationality of some p-adic L-values

We give a proof of the irrationality of the p-adic zeta-values ζ p (k) for p = 2, 3 and k = 2, 3. Such results were recently obtained by F.Calegari as an application of overconvergent p-adic modular

At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational

  • Li LaiLi Zhou
  • Mathematics, Philosophy
    Publicationes Mathematicae Debrecen
  • 2022
Let ζ(s) be the Riemann zeta function. We prove the statement in the title, which improves a recent result of Rivoal and Zudilin by lowering 69 to 35. We also prove that at least one of β(2), β(4), .

A Simple Proof that ζ(2) is Irrational

We prove that a partial sum of ζ(2) − 1 = z2 is not given by any single decimal in a number base given by a denominator of its terms. This result, applied to all partials, shows that there are an

HYPERGEOMETRIC CONSTRUCTIONS OF RATIONAL APPROXIMATIONS FOR (MULTIPLE) ZETA VALUES

This survey presents certain results concerning the diophantine nature of zeta values or multiple zeta values that I have obtained over the last few years, with or without coauthors. I did not try to
...

References

SHOWING 1-10 OF 102 REFERENCES

Irrationality of values of the Riemann zeta function

The paper deals with a generalization of Rivoal's construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function ζ(s) only at odd points. We prove

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

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

The group structure for ζ(3)

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. ..

Rational approximations to the dilogarithm

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

A note on Beukers' integral

  • M. Hata
  • Mathematics
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics
  • 1995
Abstract The aim of this note is to give a sharp lower bound for rational approximations to ζ(2) = π2/6 by using a specific Beukers' integral. Indeed, we will show that π2 has an irrationality

Cancellation of factorials

We study the arithmetic property which allows to sharpen number-theoretic estimates. Previous results on this property are, as a rule, quantitive. The application of our general qualitive theorems to

ON IRRATIONALITY OF THE VALUES OF THE FUNCTIONS $ F(x,s)$

This paper gives a proof of a theorem on the linear independence over of values of functions , where is a rational number whose numerator and denominator satisfy a certain relation.Bibliography: 1

On irrationality measures of the values of Gauss hypergeometric function

The paper gives irrationality measures for the values of some Gauss hypergeometric functions both in the archimedean andp-adic case. Further, an improvement of general results is obtained in the case

Integral identities and constructions of approximations to zeta-values

Nous presentons une construction generale de combinaisons lineaires a coefficients rationnels en les valeurs de la fonction zeta de Riemann aux entiers. Ces formes lineaires sont exprimees en termes

Generalized hypergeometric series

This also gives in the paper T. H. Koornwinder, Orthogonal polynomials with weight function (1− x)α(1 + x)β + Mδ(x + 1) + Nδ(x− 1), Canad. Math. Bull. 27 (1984), 205–214 the identitity (2.5) with N =
...