In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Pad{é} approximation problems at several points, both at functional and numerical levels (i.e., before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky theorem on values of E-functions: Shidlovsky's lemma is used to replace Nesterenko's linear independence criterion with Siegel's, so that no lower bound is needed on the linear forms in zeta values. The same strategy provides a new proof, and a refinement, of Nishimoto's theorem on values of L-functions of Dirichlet characters.
In a recent paper with Sprang and Zudilin, the following result was proved: if $a$ is large enough in terms of $\varepsilon>0$, then at least $2^{(1-\varepsilon)\frac{\log a}{\log \log a}}$ values of the Riemann zeta function at odd integers between $3$ and $a$ are irrational. This improves on the Ball-Rivoal theorem, that provides only $\frac{1-\varepsilon}{1+\log 2} \log a$ such irrational values -- but with a stronger property: they are linearly independent over the rationals.In the present paper we generalize this recent result to both $L$-functions of Dirichlet characters and Hurwitz zeta…
There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched fo…
Let $p$ be a prime number and $\mathbb{C}_p$ the completion of algebraic closure of $\mathbb{Q}_p$. Let $K$ be an algebraic number field. We fix an embedding $ι_p:\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ and denote $K_p$ the completion of $K$ with respect to the embedding $ι_p$. Let $g(z)\in K[[z]]$ and denote by $\mathcal{M}(g)(z)\in \tfrac{1}{z}K\left[\left[\tfrac{1}{z}\right]\right]$ the formal Mellin transform of $g(z)$. In this article, we prove that if $\mathcal{M}(g)(z)$ has a good Padé approximation, the special values $\mathcal{M}(g)(α)$ are convergent in $K_p$ and irrationa…
Available proofs of result of the type 'at least one of the odd zeta values $ζ(5),ζ(7),\dots,ζ(s)$ is irrational' make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are however counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result - a proof whose technical ingredients are limited to the prime numbe…
设f是一个G-函数（在Siegel意义下），x是一个代数数;假设f（x）是一个实数。作为一个更一般结果的特例，我们证明了f（x）可以写成g（1），其中g是具有任意大收敛半径的有理系数G-函数.作为应用，我们证明了这类值的乘积恰好是序列a（n）/b（n）的极限数，其中这两个序列的生成列都是有理系数的G-函数。这一结果为zeta（3）的无理性证明提供了一个一般的设置，并回答了T. Rivoal in [Approximations rationnelles des valeurs de la fongly Gamma aux rationnels：le cas des puissances，Acta Arith. 142（2010），no. 4，347-365]。
