arXiv subjects

Stéphane Fischler

Publications and source records attributed to Stéphane Fischler.

At least 19 recordsLinked to original sources

arXiv2026

Transcendence measure for the values of the logarithmic derivative of the Bessel function $J_0$ at algebraic arguments

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\varepsilon>0$, there exists $c>0$ such that $\vert P(J_0'(α)/J_0(α))\vert>c/H^{4d^2δ+\varepsilon}$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2δ$ can be replaced by a smaller (explicit) quantity $μ(d,δ)\le 4d^2δ-2dδ-1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f'$ are homogeneously algebraically independent. Our method rests upon the optimization of the size of a determinant that appears naturally in the classical Siegel-Shidlovskii method, following the steps of our previous improvement of the transcendence measure of the value $e^α$ for any $α\in \overline{\mathbb Q}^*$.

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arXiv2026

On the singularities of differential equations satisfied by $E$-functions

Let $ξ$ be a value, at an algebraic point, of a Siegel $E$-function. As a special case of a very general interpolation result, we prove that there exists an $E$-function $f$ such that $f(1)=ξ$, and such that 1 is not a singularity of the minimal differential equation satisfied by $f$. We prove that the same property does not hold at the point $0$, when $ξ$ is the value at a non-zero algebraic number of the Bessel function. This answers an analogue of a question asked by Yves Andr{é} for $G$-functions.

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arXiv2025

Linear independence of odd zeta values using Siegel's lemma

We prove that among 1 and the odd zeta values $ζ(3)$, $ζ(5)$, \ldots, $ζ(s)$, at least $ 0.21 \sqrt{s}/\sqrt{\log s}$ are linearly independent over the rationals, for any sufficiently large odd integer $s$. This is the first asymptotic improvement on the lower bound, logarithmic in $s$, obtained by Ball-Rivoal in 2001. The proof is based on Siegel's lemma to construct non-explicit linear forms in values at odd integers of the Riemann zeta function, instead of using explicit well-poised hypergeometric series. A new refinement of Siegel's linear independence criterion is applied, together with a multiplicity estimate (namely a generalization of Shidlovsky's lemma). The result is also adapted to deal with values of the first $s$ polylogarithms at a fixed algebraic point in the unit disk, improving bounds of Rivoal and Marcovecchio.

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arXiv2025

Rational approximations to values of $E$-functions

We solve a long standing problem in the theory of Siegel's $E$-functions, initiated by Lang for Bessel's function $J_0$ in the 60's and considered in full generality by G. Chudnovsky in the 80's: we prove that irrational values taken at rational points by $E$-functions with rational Taylor coefficients have irrationality exponent equal to 2. This result had been obtained before by Zudilin under strong assumptions on algebraic independence of $E$-functions, satisfied by $J_0$ but not by all hypergeometric $E$-functions for instance. We remove them using a new generalization of Shidlovskii's lemma, analogous to zero estimates on commutative algebraic groups in which obstructions come from algebraic subgroups.

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arXiv2025

Zeros of $E$-functions and of exponential polynomials defined over $\overline{\mathbb{Q}}$

Zeros of Bessel functions $J_α$ play an important role in physics. They are a motivation for studying zeros of exponential polynomials defined over $\overline{\mathbb{Q}}$, and more generally of $E$-functions. In this paper we partially characterize $E$-functions with zeros of the same multiplicity, and prove a special case of a conjecture of Jossen on entire quotients of $E$-functions, related to Ritt's theorem and Shapiro's conjecture on exponential polynomials. We also deduce from Schanuel's conjecture many results on zeros of exponential polynomials over $\overline{\mathbb{Q}}$, including $π$, logarithms of algebraic numbers, and zeros of $J_α$ when $2α$ is an odd integer. For the latter we define (if $α\neq\pm1/2$) an analogue of the minimal polynomial and Galois conjugates of algebraic numbers. At last, we study conjectural generalizations to factorization and zeros of $E$-functions.

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arXiv2025

A new transcendence measure for the values of the exponential function at algebraic arguments

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $δ\ge 1$ and usual height $H\ge 1$, and let $α\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. Mahler proved in 1931 the following transcendence measure for $e^α$: for any $\varepsilon\>0$, there exists $c\>0$ such that $\vert P(e^α)\vert\>c/H^{μ(d,δ)+\varepsilon}$ where the exponent $μ(d,δ)=(4d^2-2d)δ+2d-1$. Zheng obtained a better result in 1991 with $μ(d,δ)=(4d^2-2d)δ-1$. In this paper, we provide a new explicit exponent $μ(d,δ)$ which improves on Zheng's transcendence measure for all $δ\ge 2$ and all $d\ge 2$. When $δ=1$, we recover his bound for all $d\ge 2$, which had in fact already been obtained by Kappe in 1966. Our improvement rests upon the optimization of an accessory parameter in Siegel's classical determinant method applied to Hermite-Pad{é} approximants to powers of the exponential function.

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arXiv2024

Transcendence of values of logarithms of $E$-functions

Let $f$ be an $E$-function (in Siegel's sense) not of the form $e^{βz}$, $β\in \overline{\mathbb{Q}}$, and let $\log$ denote any fixed determination of the complex logarithm. We first prove that there exists a finite set $S(f)$ such that for all $ξ\in \overline{\mathbb{Q}}\setminus S(f)$, $\log(f(ξ))$ is a transcendental number. We then quantify this result when $f$ is an $E$-function in the strict sense with rational coefficients, by proving an irrationality measure of $\ln(f(ξ))$ when $ξ\in \mathbb{Q}\setminus S(f)$ and $f(ξ)\gt0$. This measure implies that $\ln(f(ξ))$ is not an ultra-Liouville number, as defined by Marques and Moreira. The proof of our first result, which is in fact more general, uses in particular a recent theorem of Delaygue. The proof of the second result, which is independent of the first one, is a consequence of a new linear independence measure for values of linearly independent $E$-functions in the strict sense with rational coefficients, where emphasis is put on other parameters than on the height, contrary to the case in Shidlovskii's classical measure for instance.

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arXiv2023

Values of E-functions are not Liouville numbers

Shidlovskii has given a linear independence measure of values of $E$-functions with rational Taylor coefficients at a rational point, not a singularity of the underlying differential system satisfied by these $E$-functions. Recently, Beukers has proved a qualitative linear independence theorem for the values at an algebraic point of $E$-functions with arbitrary algebraic Taylor coefficients. In this paper, we obtain an analogue of Shidlovskii's measure for values of arbitrary $E$-functions at algebraic points. This enables us to solve a long standing problem by proving that the value of an $E$-function at an algebraic point is never a Liouville number. We also prove that values at rational points of $E$-functions with rational Taylor coefficients are linearly independent over $\overline{\mathbb{Q}}$ if and only if they are linearly independent over $\mathbb{Q}$. Our methods rest upon improvements of results obtained by André and Beukers in the theory of $E$-operators.

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arXiv2023

Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function

We investigate the relations between the rings ${\bf E}$, ${\bf G}$ and ${\bf D}$ of values taken at algebraic points by arithmetic Gevrey series of order either $-1$ ($E$-functions), $0$ (analytic continuations of $G$-functions) or $1$ (renormalization of divergent series solutions at $\infty$ of $E$-operators) respectively. We prove in particular that any element of ${\bf G}$ can be written as multivariate polynomial with algebraic coefficients in elements of ${\bf E}$ and ${\bf D}$, and is the limit at infinity of some $E$-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously $E$-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for $E$-functions) and the conjecture ${\bf D}\cap{\bf E}=\overline{\mathbb Q}$ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of $E$-functions, the transcendance of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendance of Gompertz constant as well as the fact that Euler's constant is not in ${\bf E}$.

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arXiv2021

Filling times for linear flow on the torus with truncated Diophantine conditions: a brief review and new proof

We show that the geometry-of-numbers method used by A. Bounemoura to obtain filling times for linear flow on the torus satisfying Diophantine conditions may be extended to the case of linear flow with truncated Diophantine conditions, and we use these methods to recover the optimal estimate first obtained by M. Berti, L. Biasco, and P. Bolle in 2003. We also briefly review the dynamics of linear flow on the torus, previous results, optimality, and applications of these estimates.

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arXiv2021

A note on G-operators of order 2

It is known that $G$-functions solutions of a linear differential equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a $G$-function solution of an inhomogeneous equation of order 1 with coefficients in $\overline{\mathbb{Q}}(z)$, as well as that of a $G$-function $f$ of differential order 2 over $\overline{\mathbb{Q}}(z)$, and such that $f$ and $f'$ are algebraically dependent over $\mathbb{C}(z)$. Our results apply more generally to Nilsson-Gevrey arithmetic series of order 0 that encompass $G$-functions.

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arXiv2019

Irrationality of 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 function. The strategy is different and less elementary: the construction is related to a Padé approximation problem, and a generalization of Shidlovsky's lemma is used to apply Siegel's linear independence criterion. We also improve the analogue of the Ball-Rivoal theorem in this setting: we obtain $\frac{1-\varepsilon}{1+\log 2} \log a$ linearly independent values $L(f,s)$ with $s\leq a$ of a fixed parity, when $f$ is a Dirichlet character. The new point here is that the constant $1+\log 2$ does not depend on $f$.

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arXiv2018

Linear independence of values of G-functions, II. Outside the disk of convergence

Given any non-polynomial $G$-function $F(z)=\sum_{k=0}^\infty A_k z^k$ of radius of convergence $R$ and in the kernel a $G$-operator $L_F$, we consider the $G$-functions $F_n^{[s]}(z)=\sum_{k=0}^\infty \frac{A_k}{(k+n)^s}z^k$ for every integers $s\ge 0$ and $n\ge 1$. These functions can be analytically continued to a domain $\mathcal{D}_F$ star-shaped at $0$ and containing the disk $\{\vert z\vert 0$ and $v_F>0$. This appears to be the first Diophantine result for values of $G$-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [{\em Linear independence of values of G-functions}, 46 pages, J. Europ. Math. Soc., to appear], where $α\in \overline{\mathbb{Q}}^*$ was assumed to be such that $\vert α\vert <R$. Its proof relies on an explicit construction of a Padé approximation problem adapted to certain non-holomorphic functions associated to $F$, and it is quite different of that in the above mentioned paper. It makes use of results of André, Chudnovsky and Katz on $G$-operators, of a linear independence criterion à la Siegel over number fields, and of a far reaching generalization of Shidlovsky's lemma built upon the approach of Bertrand-Beukers and Bertrand.

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arXiv2018

Many odd zeta values are irrational

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any positive real number and $s$ is large enough in terms of $\varepsilon$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $\frac{1-\varepsilon}{1+\log2}\log s$ that follows from the Ball--Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

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arXiv2017

Linear independence of values of G-functions

Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For any fixed algebraic number $α$ such that $0 \textless{} \vert α\vert \textless{} R$ and any number field $\mathbb{K}$ containing $α$ and the $A\_k$'s, we define $Φ\_{α, S}$ as the $\mathbb{K}$-vector space generated by the values $F\_n^{[s]}(α)$, $n\ge 1$ and $0\leq s\leq S$. We prove that $u\_{\mathbb{K},F}\log(S)\leq \dim\_{\mathbb{K}}(Φ\_{α, S})\leq v\_F S$ for any $S$, with effective constants $u\_{\mathbb{K},F}\textgreater{}0$ and $v\_F\textgreater{}0$, and that the family $(F\_n^{[s]}(α))\_{1\le n \le v\_F, s \ge 0}$ contains infinitely many irrational numbers. This theorem applies in particular when $F$ is an hypergeometric series with rational parameters or a multiple polylogarithm, and it encompasses a previous result by the second author and Marcovecchio in the case of polylogarithms. The proof relies on an explicit construction of Padé-type approximants. It makes use of results of André, Chudnovsky and Katz on $G$-operators, of a new linear independence criterion à la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.

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arXiv2016

Shidlovsky's multiplicity estimate and Irrationality of zeta values

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.

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arXiv2015

Microsolutions of differential operators and values of arithmetic Gevrey series

We continue our investigation of $E$-operators, in particular their connection with $G$-operators; these differential operators are fundamental in understanding the diophantine properties of Siegel's $E$ and $G$-functions. We study in detail microsolutions (in Kashiwara's sense) of Fuchsian differential operators, and apply this to the construction of basis of solutions at $0$ and $\infty$ of any $E$-operator from microsolutions of a $G$-operator; this provides a constructive proof of a theorem of André. We also focus on the arithmetic nature of connection constants and Stokes constants between different bases of solutions of $E$-operators. For this, we introduce and study in details an arithmetic (inverse) Laplace transform that enables one to get rid of transcendental numbers inherent to André's original approach. As an application, we define a set of special values of arithmetic Gevrey series, and discuss its conjectural relation with the ring of exponential periods.

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arXiv2013

Distribution of irrational zeta values

In this paper we refine Ball-Rivoal's theorem by proving that for any odd integer $a$ sufficiently large in terms of $ε>0$, there exist $[ \frac{(1-ε)\log a}{1+\log 2}]$ odd integers $s$ between 3 and $a$, with distance at least $a^ε$ from one another, at which Riemann zeta function takes $\Q$-linearly independent values. As a consequence, if there are very few integers $s$ such that $ζ(s)$ is irrational, then they are rather evenly distributed. The proof involves series of hypergeometric type estimated by the saddle point method, and the generalization to vectors of Nesterenko's linear independence criterion.

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