{"title":"算术格弗里数列值之间的关系,以及对伽马函数值的应用","authors":"S. Fischler , T. Rivoal","doi":"10.1016/j.jnt.2024.02.016","DOIUrl":null,"url":null,"abstract":"<div><p>We investigate the relations between the rings <strong>E</strong>, <strong>G</strong> and <strong>D</strong> of values taken at algebraic points by arithmetic Gevrey series of order either −1 (<em>E</em>-functions), 0 (analytic continuations of <em>G</em>-functions) or 1 (renormalization of divergent series solutions at ∞ of <em>E</em>-operators) respectively. We prove in particular that any element of <strong>G</strong> can be written as multivariate polynomial with algebraic coefficients in elements of <strong>E</strong> and <strong>D</strong>, and is the limit at infinity of some <em>E</em>-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously <em>E</em>-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 <em>E</em>-functions) and the conjecture <span><math><mi>D</mi><mo>∩</mo><mi>E</mi><mo>=</mo><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></math></span> (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 <em>E</em>-functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in <strong>E</strong>.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"261 ","pages":"Pages 36-54"},"PeriodicalIF":0.6000,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function\",\"authors\":\"S. Fischler , T. Rivoal\",\"doi\":\"10.1016/j.jnt.2024.02.016\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We investigate the relations between the rings <strong>E</strong>, <strong>G</strong> and <strong>D</strong> of values taken at algebraic points by arithmetic Gevrey series of order either −1 (<em>E</em>-functions), 0 (analytic continuations of <em>G</em>-functions) or 1 (renormalization of divergent series solutions at ∞ of <em>E</em>-operators) respectively. We prove in particular that any element of <strong>G</strong> can be written as multivariate polynomial with algebraic coefficients in elements of <strong>E</strong> and <strong>D</strong>, and is the limit at infinity of some <em>E</em>-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously <em>E</em>-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 <em>E</em>-functions) and the conjecture <span><math><mi>D</mi><mo>∩</mo><mi>E</mi><mo>=</mo><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></math></span> (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 <em>E</em>-functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in <strong>E</strong>.</p></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"261 \",\"pages\":\"Pages 36-54\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-03-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X2400057X\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X2400057X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function
We investigate the relations between the rings E, G and 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 ∞ of E-operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and 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 (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 transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.