# 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} }

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:

## 97 Citations

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

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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.

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

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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

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- 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

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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

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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…

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