Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2210.12046

A Lindemann-Weierstrass theorem for $E$-functions

É. Delaygue
View original source ↗Read source document ↗Export metadata ↓

Abstract

$E$-functions were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function. After developments of Siegel's methods by Shidlovskii, Nesterenko and André, Beukers proved in 2006 an optimal result on the algebraic independence of the values of $E$-functions which generalizes the Lindemann-Weierstrass theorem. Since then, it seems that no general result was stated concerning the relations between the values of a single $E$-function. We prove that André's theory of $E$-operators and Beuker's result lead to a Lindemann-Weierstrass theorem for $E$-functions in its linear independence formulation. As a consequence, we show that all transcendental values at algebraic arguments of an entire hypergeometric function are linearly independent over $\overline{\mathbb{Q}}$.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

É. Delaygue. 2025-03-07. A Lindemann-Weierstrass theorem for $E$-functions. https://arxiv.org/abs/2210.12046

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT↗
arXiv2026

Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

math.NT↗