Transcendence of values of logarithms of $E$-functions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910622941184000 |
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| author | Fischler, Stéphane Rivoal, Tanguy |
| author_facet | Fischler, Stéphane Rivoal, Tanguy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18537 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transcendence of values of logarithms of $E$-functions Fischler, Stéphane Rivoal, Tanguy Number Theory 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. |
| title | Transcendence of values of logarithms of $E$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.18537 |