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Weak almost monomial groups and Artin's conjecture 弱几乎单项式群和阿尔丁猜想
arXiv - MATH - Number Theory Pub Date : 2024-09-09 DOI: arxiv-2409.05629
Mircea Cimpoeas
{"title":"Weak almost monomial groups and Artin's conjecture","authors":"Mircea Cimpoeas","doi":"arxiv-2409.05629","DOIUrl":"https://doi.org/arxiv-2409.05629","url":null,"abstract":"We introduce a new class of finite groups, called weak almost monomial, which\u0000generalize two different notions of \"almost monomial\" groups, and we prove it\u0000is closed under taking factor groups and direct products. Let $K/mathbb Q$ be a finite Galois extension with a weak almost monomial\u0000Galois group $G$ and $s_0in mathbb Csetminus {1}$. We prove that Artin\u0000conjecture's is true at $s_0$ if and only if the monoid of holomorphic Artin\u0000$L$-functions at $s_0$ is factorial. Also, we show that if $s_0$ is a simple\u0000zero for some Artin $L$-function associated to an irreducible character of $G$\u0000and it is not a zero for any other $L$-function associated to an irreducible\u0000character, then Artin conjecture's is true at $s_0$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Plateaux of probability for the expanded quantum infinite well 扩展量子无限井的概率高原
arXiv - MATH - Number Theory Pub Date : 2024-09-09 DOI: arxiv-2409.06058
Fernando Chamizo, Dulcinea Raboso, Osvaldo P. Santillán
{"title":"Plateaux of probability for the expanded quantum infinite well","authors":"Fernando Chamizo, Dulcinea Raboso, Osvaldo P. Santillán","doi":"arxiv-2409.06058","DOIUrl":"https://doi.org/arxiv-2409.06058","url":null,"abstract":"If the standard 1D quantum infinite potential well initially in its ground\u0000state suffers a sudden expansion, it turns out that in the evolution of the\u0000system they may appear plateaux of probability for some fractional times, as\u0000noticed by C. Aslangul in 2008. We introduce a mathematical framework to\u0000explain this phenomenon. Remarkably, the characterization of these plateaux\u0000depends on nontrivial number theoretical considerations.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"408 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Constructions of well-rounded algebraic lattices over odd prime degree cyclic number fields 奇素数域上完备代数网格的构建
arXiv - MATH - Number Theory Pub Date : 2024-09-07 DOI: arxiv-2409.04839
Robson Ricardo de Araujo, Antônio Aparecido de Andrade, Trajano Pires da Nóbrega Neto, Jéfferson Luiz Rocha Bastos
{"title":"Constructions of well-rounded algebraic lattices over odd prime degree cyclic number fields","authors":"Robson Ricardo de Araujo, Antônio Aparecido de Andrade, Trajano Pires da Nóbrega Neto, Jéfferson Luiz Rocha Bastos","doi":"arxiv-2409.04839","DOIUrl":"https://doi.org/arxiv-2409.04839","url":null,"abstract":"Algebraic lattices are those obtained from modules in the ring of integers of\u0000algebraic number fields through the canonical or twisted embeddings. In turn,\u0000well-rounded lattices are those with maximal cardinality of linearly\u0000independent vectors in its set of minimal vectors. Both classes of lattices\u0000have been applied for signal transmission in some channels, such as wiretap\u0000channels. Recently, some advances have been made in the search for well-rounded\u0000lattices that can be realized as algebraic lattices. Moreover, some works have\u0000been published studying algebraic lattices obtained from modules in cyclic\u0000number fields of odd prime degree $p$. In this work, we generalize some results\u0000of a recent work of Tran et al. and we provide new constructions of\u0000well-rounded algebraic lattices from a certain family of modules in the ring of\u0000integers of each of these fields when $p$ is ramified in its extension over the\u0000field of rational numbers.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"102 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sato tau functions and construction of Somos sequence 佐藤头函数和索莫斯序列的构建
arXiv - MATH - Number Theory Pub Date : 2024-09-07 DOI: arxiv-2409.05911
Mohamed Bensaid
{"title":"Sato tau functions and construction of Somos sequence","authors":"Mohamed Bensaid","doi":"arxiv-2409.05911","DOIUrl":"https://doi.org/arxiv-2409.05911","url":null,"abstract":"In this short article, we will reconstruct the KP equation from Plucker\u0000relations and provide some generalizations on this topic. Additionally, in the\u0000final section, we define the discrete function $tau$ in a similar manner,\u0000leading to the construction of an integer sequence that has not yet been listed\u0000in the OEIS. Furthermore, this approach allows us to construct many other\u0000sequences that are not listed in the OEIS.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Ramsey-type problems for generalised Sidon sets 广义西顿集合的拉姆齐型问题
arXiv - MATH - Number Theory Pub Date : 2024-09-07 DOI: arxiv-2409.04809
Vojtěch Rödl, Christian Reiher, Mathias Schacht
{"title":"Ramsey-type problems for generalised Sidon sets","authors":"Vojtěch Rödl, Christian Reiher, Mathias Schacht","doi":"arxiv-2409.04809","DOIUrl":"https://doi.org/arxiv-2409.04809","url":null,"abstract":"We establish the existence of generalised Sidon sets enjoying additional\u0000Ramsey-type properties, which are motivated by questions of ErdH{o}s and\u0000Newman and of Alon and ErdH{o}s.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"408 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lattice point counting statistics for 3-dimensional shrinking Cygan-Korányi spherical shells 三维收缩 Cygan-Korányi 球壳的晶格点计数统计
arXiv - MATH - Number Theory Pub Date : 2024-09-07 DOI: arxiv-2409.04814
Yoav A. Gath
{"title":"Lattice point counting statistics for 3-dimensional shrinking Cygan-Korányi spherical shells","authors":"Yoav A. Gath","doi":"arxiv-2409.04814","DOIUrl":"https://doi.org/arxiv-2409.04814","url":null,"abstract":"Let $E(x;omega)$ be the error term for the number of integer lattice points\u0000lying inside a $3$-dimensional Cygan-Kor'anyi spherical shell of inner radius\u0000$x$ and gap width $omega(x)>0$. Assuming that $omega(x)to0$ as $xtoinfty$,\u0000and that $omega$ satisfies suitable regularity conditions, we prove that\u0000$E(x;omega)$, properly normalized, has a limiting distribution. Moreover, we\u0000show that the corresponding distribution is moment-determinate, and we give a\u0000closed form expression for its moments. As a corollary, we deduce that the\u0000limiting distribution is the standard Gaussian measure whenever $omega$ is\u0000slowly varying. We also construct gap width functions $omega$, whose\u0000corresponding error term has a limiting distribution that is absolutely\u0000continuous with a non-Gaussian density.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bessenrodt--Ono inequalities for $ell$-tuples of pairwise commuting permutations 成对换向排列的 $ell$ 元组的贝森罗德--奥诺不等式
arXiv - MATH - Number Theory Pub Date : 2024-09-07 DOI: arxiv-2409.04881
Abdelmalek Abdesselam, Bernhard Heim, Markus Neuhauser
{"title":"Bessenrodt--Ono inequalities for $ell$-tuples of pairwise commuting permutations","authors":"Abdelmalek Abdesselam, Bernhard Heim, Markus Neuhauser","doi":"arxiv-2409.04881","DOIUrl":"https://doi.org/arxiv-2409.04881","url":null,"abstract":"Let $S_n$ denote the symmetric group. We consider begin{equation*}\u0000N_{ell}(n) := frac{leftvert Homleft( mathbb{Z}^{ell},S_nright)\u0000rightvert}{n!} end{equation*} which also counts the number of $ell$-tuples\u0000$pi=left( pi_1, ldots, pi_{ell}right) in S_n^{ell}$ with $pi_i pi_j\u0000= pi_j pi_i$ for $1 leq i,j leq ell$ scaled by $n!$. A recursion formula,\u0000generating function, and Euler product have been discovered by Dey, Wohlfahrt,\u0000Bryman and Fulman, and White. Let $a,b, ell geq 2$. It is known by Bringman,\u0000Franke, and Heim, that the Bessenrodt--Ono inequality begin{equation*}\u0000Delta_{a,b}^{ell}:= N_{ell}(a) , N_{ell}(b) - N_{ell}(a+b) >0\u0000end{equation*} is valid for $a,b gg 1$ and by Bessenrodt and Ono that it is\u0000valid for $ell =2$ and $a+b >9$. In this paper we prove that for each pair\u0000$(a,b)$ the sign of ${Delta_{a,b}^{ell} }_{ell}$ is getting stable. In\u0000each case we provide an explicit bound. The numbers $N_{ell}left( nright) $\u0000had been identified by Bryan and Fulman as the $n$-th orbifold characteristics,\u0000generalizing work by Macdonald and Hirzebruch--H\"{o}fer concerning the\u0000ordinary and string-theoretic Euler characteristics of symmetric products,\u0000where $N_2(n)=p(n) $ represents the partition function.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Primes in Tuples of Linear Forms in Number Fields and Function Fields 数域和函数域中线性形式元组中的素数
arXiv - MATH - Number Theory Pub Date : 2024-09-07 DOI: arxiv-2409.04705
Habibur Rahaman
{"title":"Primes in Tuples of Linear Forms in Number Fields and Function Fields","authors":"Habibur Rahaman","doi":"arxiv-2409.04705","DOIUrl":"https://doi.org/arxiv-2409.04705","url":null,"abstract":"Following the work of Castillo-Hall-Oliver-Pollack-Thompson who extended\u0000Maynard-Tao theorem on admissible tuples to number fields and function fields\u0000for tuples with monic linear forms, here we obtain the Maynard-Tao theorem for\u0000admissible tuples of linear forms with arbitrary leading coefficients in number\u0000fields and function fields. Also, we provide some applications of our results.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
L-Series for Vector-Valued Weakly Holomorphic Modular Forms and Converse Theorems 矢量值弱全态模态的 L 序列和逆定理
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04258
Subong Lim, Wissam Raji
{"title":"L-Series for Vector-Valued Weakly Holomorphic Modular Forms and Converse Theorems","authors":"Subong Lim, Wissam Raji","doi":"arxiv-2409.04258","DOIUrl":"https://doi.org/arxiv-2409.04258","url":null,"abstract":"We introduce the $L$-series of weakly holomorphic modular forms using Laplace\u0000transforms and give their functional equations. We then determine converse\u0000theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and\u0000elliptic modular forms of half-integer weight in Kohnen plus space.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"103 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extending a result of Carlitz and McConnel to polynomials which are not permutations 将 Carlitz 和 McConnel 的一个结果扩展到非排列的多项式
arXiv - MATH - Number Theory Pub Date : 2024-09-06 DOI: arxiv-2409.04045
Bence Csajbók
{"title":"Extending a result of Carlitz and McConnel to polynomials which are not permutations","authors":"Bence Csajbók","doi":"arxiv-2409.04045","DOIUrl":"https://doi.org/arxiv-2409.04045","url":null,"abstract":"Let $D$ denote the set of directions determined by the graph of a polynomial\u0000$f$ of $mathbb{F}_q[x]$, where $q$ is a power of the prime $p$. If $D$ is\u0000contained in a multiplicative subgroup $M$ of $mathbb{F}_q^times$, then by a\u0000result of Carlitz and McConnel it follows that $f(x)=ax^{p^k}+b$ for some $kin\u0000mathbb{N}$. Of course, if $Dsubseteq M$, then $0notin D$ and hence $f$ is a\u0000permutation. If we assume the weaker condition $Dsubseteq M cup {0}$, then\u0000$f$ is not necessarily a permutation, but Sziklai conjectured that\u0000$f(x)=ax^{p^k}+b$ follows also in this case. When $q$ is odd, and the index of\u0000$M$ is even, then a result of Ball, Blokhuis, Brouwer, Storme and SzH onyi\u0000combined with a result of McGuire and G\"olou{g}lu proves the conjecture.\u0000Assume $deg fgeq 1$. We prove that if the size of $D^{-1}D={d^{-1}d' : din\u0000Dsetminus {0},, d'in D}$ is less than $q-deg f+2$, then $f$ is a\u0000permutation of $mathbb{F}_q$. We use this result to verify the conjecture of\u0000Sziklai.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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