Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hyperg{\'e}om{\'e}trie et fonction z{\^e}ta de Riemann   We prove the second author's "denominator conjecture" [40] concerning the
common denominators of coefficients of certain linear forms in zeta values.
These forms were recently constructed to obtain lower bounds for the dimension
of the vector space over $\mathbb Q$ spanned by
$1,\zeta(m),\zeta(m+2),...,\zeta(m+2h)$, where $m$ and $h$ are integers such
that $m\ge2$ and $h\ge0$. In particular, we immediately get the following
results as corollaries: at least one of the eight numbers
$\zeta(5),\zeta(7),...,\zeta(19)$ is irrational, and there exists an odd
integer $j$ between 5 and 165 such that 1, $\zeta(3)$ and $\zeta(j)$ are
linearly independent over $\mathbb{Q}$. This strengthens some recent results in
[41] and [8], respectively. We also prove a related conjecture, due to Vasilyev
[49], and as well a conjecture, due to Zudilin [55], on certain rational
approximations of $\zeta(4)$. The proofs are based on a hypergeometric identity
between a single sum and a multiple sum due to Andrews [3]. We hope that it
will be possible to apply our construction to the more general linear forms
constructed by Zudilin [56], with the ultimate goal of strengthening his result
that one of the numbers $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational.

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The categories of this paper are: q-alg
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: astro-ph