. In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
{"title":"Combinatorial aspects of poly-Bernoulli polynomials and poly-Euler numbers","authors":"Be'ata B'enyi, Toshiki Matsusaka","doi":"10.5802/jtnb.1234","DOIUrl":"https://doi.org/10.5802/jtnb.1234","url":null,"abstract":". In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46372866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any non-integral positive real number c, any sequence (bnc)n is called a Pjateckii-Šapiro sequence. Given a real number c in the interval ( 1, 12 11 ) , it is known that the number of primes in this sequence up to x has an asymptotic formula. We would like to use the techniques of Gupta and Murty to study Artin’s problems for such primes. We will prove that even though the set of Pjateckii-Šapiro primes is of density zero for a fixed c, one can show that there exist natural numbers which are primitive roots for infinitely many Pjateckii-Šapiro primes for any fixed c in the interval ( 1, √ 77 7 − 1 4 ) .
{"title":"Primitive roots for Pjateckii-Šapiro primes","authors":"J. Sivaraman","doi":"10.5802/JTNB.1152","DOIUrl":"https://doi.org/10.5802/JTNB.1152","url":null,"abstract":"For any non-integral positive real number c, any sequence (bnc)n is called a Pjateckii-Šapiro sequence. Given a real number c in the interval ( 1, 12 11 ) , it is known that the number of primes in this sequence up to x has an asymptotic formula. We would like to use the techniques of Gupta and Murty to study Artin’s problems for such primes. We will prove that even though the set of Pjateckii-Šapiro primes is of density zero for a fixed c, one can show that there exist natural numbers which are primitive roots for infinitely many Pjateckii-Šapiro primes for any fixed c in the interval ( 1, √ 77 7 − 1 4 ) .","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75170536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $ain mathbb{Z}$ and $rho$ be a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, then the number field $K_a=mathbb{Q}(rho)$ is called a simplest cubic field. In this paper we consider the family of unit equations $u_1+u_2=n$ where $u_1,u_2in mathbb{Z}[rho]^*$ and $nin mathbb{Z}$. We completely solve the unit equations under the restriction $|n|leq max{1,|a|^{1/3}}$.
{"title":"On a family of unit equations over simplest cubic fields","authors":"I. Vukusic, V. Ziegler","doi":"10.5802/jtnb.1223","DOIUrl":"https://doi.org/10.5802/jtnb.1223","url":null,"abstract":"Let $ain mathbb{Z}$ and $rho$ be a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, then the number field $K_a=mathbb{Q}(rho)$ is called a simplest cubic field. In this paper we consider the family of unit equations $u_1+u_2=n$ where $u_1,u_2in mathbb{Z}[rho]^*$ and $nin mathbb{Z}$. We completely solve the unit equations under the restriction $|n|leq max{1,|a|^{1/3}}$.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49253689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Following Humbert and Lagarias, given a real number θ, we call a nonzero vector (p, q) ∈ Z × N a Hermite best approximation vector of θ if it minimizes a quadratic form f∆(x, y) = (x− yθ)2 + y 2 ∆ for at least one real number ∆ > 0. Hermite observed that if (p, q) is such a minimum with q > 0, then the fraction p/q must be a convergent of the continued fraction expansion of θ. Using minimal vectors in lattices, we give new proofs of some results of Humbert and Meignen and complete their works. In particular, we show that the proportion of Hermite best approximation vectors among convergents is almost surely ln 3/ ln 4. The main tool of the proofs is the natural extension of the Gauss map x ∈ ]0, 1[→ {1/x}.
{"title":"The natural extension of the Gauss map and the Hermite best approximations","authors":"N. Chevallier","doi":"10.5802/jtnb.1219","DOIUrl":"https://doi.org/10.5802/jtnb.1219","url":null,"abstract":"Following Humbert and Lagarias, given a real number θ, we call a nonzero vector (p, q) ∈ Z × N a Hermite best approximation vector of θ if it minimizes a quadratic form f∆(x, y) = (x− yθ)2 + y 2 ∆ for at least one real number ∆ > 0. Hermite observed that if (p, q) is such a minimum with q > 0, then the fraction p/q must be a convergent of the continued fraction expansion of θ. Using minimal vectors in lattices, we give new proofs of some results of Humbert and Meignen and complete their works. In particular, we show that the proportion of Hermite best approximation vectors among convergents is almost surely ln 3/ ln 4. The main tool of the proofs is the natural extension of the Gauss map x ∈ ]0, 1[→ {1/x}.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43378632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to p-adic L-functions under a conjectural formula for the Fitting ideals of some equivariant congruence modules for abelian base change.
{"title":"On Euler systems for adjoint Hilbert modular Galois representations","authors":"E. Urban","doi":"10.5802/jtnb.1191","DOIUrl":"https://doi.org/10.5802/jtnb.1191","url":null,"abstract":"We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to p-adic L-functions under a conjectural formula for the Fitting ideals of some equivariant congruence modules for abelian base change.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45811940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given an integer $gammageq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $mathbb{F}_q$ of genus $g$ and gonality $gamma$ and with exactly $gamma(q+1)$ $mathbb{F}_q$-rational points. This is the maximal number of rational points possible. This answers a recent conjecture by Faber--Grantham. Our methods are based on curves on toric surfaces and Poonen's work on squarefree values of polynomials.
{"title":"Curves of fixed gonality with many rational points","authors":"F. Vermeulen","doi":"10.5802/jtnb.1240","DOIUrl":"https://doi.org/10.5802/jtnb.1240","url":null,"abstract":"Given an integer $gammageq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $mathbb{F}_q$ of genus $g$ and gonality $gamma$ and with exactly $gamma(q+1)$ $mathbb{F}_q$-rational points. This is the maximal number of rational points possible. This answers a recent conjecture by Faber--Grantham. Our methods are based on curves on toric surfaces and Poonen's work on squarefree values of polynomials.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42989226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It follows from generalities of quadratic forms that the spinor class of the integral trace of a number field determines the signature and the discriminant of the field. In this paper we define a family of number fields, that contains among others all odd degree Galois tame number fields, for which the converse is true. In other words, for a number field K in such family we prove that the spinor class of the integral trace carries no more information about K than the discriminant and the signature do.
{"title":"An introduction to oddly tame number fields.","authors":"Guillermo Mantilla-Soler","doi":"10.5802/JTNB.1140","DOIUrl":"https://doi.org/10.5802/JTNB.1140","url":null,"abstract":"It follows from generalities of quadratic forms that the spinor class of the integral trace of a number field determines the signature and the discriminant of the field. In this paper we define a family of number fields, that contains among others all odd degree Galois tame number fields, for which the converse is true. In other words, for a number field K in such family we prove that the spinor class of the integral trace carries no more information about K than the discriminant and the signature do.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85947648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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{"title":"On Selberg’s Central Limit Theorem for Dirichlet L-functions","authors":"Po-Han Hsu, PENG-JIE Wong","doi":"10.5802/JTNB.1139","DOIUrl":"https://doi.org/10.5802/JTNB.1139","url":null,"abstract":"L’accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90876778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by a question of M. J. Bertin, we obtain parametrizations of minimal polynomials of quartic Salem numbers, say α, which are Mahler measures of non-reciprocal 2-Pisot numbers. This allows us to determine all such numbers α with a given trace, and to deduce that for any natural number t (resp. t ≥ 2) there is a quartic Salem number of trace t which is (resp. which is not) a Mahler measure of a non-reciprocal 2-Pisot number.
在M. J. Bertin问题的激励下，我们得到了四次Salem数的最小多项式的参数化，如α，它是非互反2-Pisot数的Mahler测度。这允许我们用给定的迹来确定所有这样的数α，并推导出对于任何自然数t (p。t≥2时，轨迹t有一个四次塞勒姆数，即(p。它不是非倒数2-皮索数的马勒测度。
{"title":"Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers","authors":"Toufik Zaïmi","doi":"10.5802/JTNB.1145","DOIUrl":"https://doi.org/10.5802/JTNB.1145","url":null,"abstract":"Motivated by a question of M. J. Bertin, we obtain parametrizations of minimal polynomials of quartic Salem numbers, say α, which are Mahler measures of non-reciprocal 2-Pisot numbers. This allows us to determine all such numbers α with a given trace, and to deduce that for any natural number t (resp. t ≥ 2) there is a quartic Salem number of trace t which is (resp. which is not) a Mahler measure of a non-reciprocal 2-Pisot number.","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84217964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Petersson norms of Eisenstein series and Kohnen–Zagier’s formula","authors":"Y. Mizuno","doi":"10.5802/JTNB.1138","DOIUrl":"https://doi.org/10.5802/JTNB.1138","url":null,"abstract":"","PeriodicalId":48896,"journal":{"name":"Journal De Theorie Des Nombres De Bordeaux","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75124743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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