We introduce a new class of finite groups, called weak almost monomial, which generalize two different notions of "almost monomial" groups, and we prove it is closed under taking factor groups and direct products. Let $K/mathbb Q$ be a finite Galois extension with a weak almost monomial Galois group $G$ and $s_0in mathbb Csetminus {1}$. We prove that Artin conjecture's is true at $s_0$ if and only if the monoid of holomorphic Artin $L$-functions at $s_0$ is factorial. Also, we show that if $s_0$ is a simple zero for some Artin $L$-function associated to an irreducible character of $G$ and it is not a zero for any other $L$-function associated to an irreducible character, then Artin conjecture's is true at $s_0$.
{"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}
Fernando Chamizo, Dulcinea Raboso, Osvaldo P. Santillán
If the standard 1D quantum infinite potential well initially in its ground state suffers a sudden expansion, it turns out that in the evolution of the system they may appear plateaux of probability for some fractional times, as noticed by C. Aslangul in 2008. We introduce a mathematical framework to explain this phenomenon. Remarkably, the characterization of these plateaux depends on nontrivial number theoretical considerations.
{"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}
Robson Ricardo de Araujo, Antônio Aparecido de Andrade, Trajano Pires da Nóbrega Neto, Jéfferson Luiz Rocha Bastos
Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent vectors in its set of minimal vectors. Both classes of lattices have been applied for signal transmission in some channels, such as wiretap channels. Recently, some advances have been made in the search for well-rounded lattices that can be realized as algebraic lattices. Moreover, some works have been published studying algebraic lattices obtained from modules in cyclic number fields of odd prime degree $p$. In this work, we generalize some results of a recent work of Tran et al. and we provide new constructions of well-rounded algebraic lattices from a certain family of modules in the ring of integers of each of these fields when $p$ is ramified in its extension over the field of rational numbers.
{"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}
In this short article, we will reconstruct the KP equation from Plucker relations and provide some generalizations on this topic. Additionally, in the final section, we define the discrete function $tau$ in a similar manner, leading to the construction of an integer sequence that has not yet been listed in the OEIS. Furthermore, this approach allows us to construct many other sequences that are not listed in the OEIS.
{"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}
We establish the existence of generalised Sidon sets enjoying additional Ramsey-type properties, which are motivated by questions of ErdH{o}s and Newman and of Alon and ErdH{o}s.
{"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}
Let $E(x;omega)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Kor'anyi spherical shell of inner radius $x$ and gap width $omega(x)>0$. Assuming that $omega(x)to0$ as $xtoinfty$, and that $omega$ satisfies suitable regularity conditions, we prove that $E(x;omega)$, properly normalized, has a limiting distribution. Moreover, we show that the corresponding distribution is moment-determinate, and we give a closed form expression for its moments. As a corollary, we deduce that the limiting distribution is the standard Gaussian measure whenever $omega$ is slowly varying. We also construct gap width functions $omega$, whose corresponding error term has a limiting distribution that is absolutely continuous with a non-Gaussian density.
{"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}
Abdelmalek Abdesselam, Bernhard Heim, Markus Neuhauser
Let $S_n$ denote the symmetric group. We consider begin{equation*} N_{ell}(n) := frac{leftvert Homleft( mathbb{Z}^{ell},S_nright) rightvert}{n!} end{equation*} which also counts the number of $ell$-tuples $pi=left( pi_1, ldots, pi_{ell}right) in S_n^{ell}$ with $pi_i pi_j = pi_j pi_i$ for $1 leq i,j leq ell$ scaled by $n!$. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryman and Fulman, and White. Let $a,b, ell geq 2$. It is known by Bringman, Franke, and Heim, that the Bessenrodt--Ono inequality begin{equation*} Delta_{a,b}^{ell}:= N_{ell}(a) , N_{ell}(b) - N_{ell}(a+b) >0 end{equation*} is valid for $a,b gg 1$ and by Bessenrodt and Ono that it is valid for $ell =2$ and $a+b >9$. In this paper we prove that for each pair $(a,b)$ the sign of ${Delta_{a,b}^{ell} }_{ell}$ is getting stable. In each case we provide an explicit bound. The numbers $N_{ell}left( nright) $ had been identified by Bryan and Fulman as the $n$-th orbifold characteristics, generalizing work by Macdonald and Hirzebruch--H"{o}fer concerning the ordinary and string-theoretic Euler characteristics of symmetric products, where $N_2(n)=p(n) $ represents the partition function.
{"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}
Following the work of Castillo-Hall-Oliver-Pollack-Thompson who extended Maynard-Tao theorem on admissible tuples to number fields and function fields for tuples with monic linear forms, here we obtain the Maynard-Tao theorem for admissible tuples of linear forms with arbitrary leading coefficients in number fields and function fields. Also, we provide some applications of our results.
{"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}
We introduce the $L$-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integer weight in Kohnen plus space.
{"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}
Let $D$ denote the set of directions determined by the graph of a polynomial $f$ of $mathbb{F}_q[x]$, where $q$ is a power of the prime $p$. If $D$ is contained in a multiplicative subgroup $M$ of $mathbb{F}_q^times$, then by a result of Carlitz and McConnel it follows that $f(x)=ax^{p^k}+b$ for some $kin mathbb{N}$. Of course, if $Dsubseteq M$, then $0notin D$ and hence $f$ is a permutation. If we assume the weaker condition $Dsubseteq M cup {0}$, then $f$ is not necessarily a permutation, but Sziklai conjectured that $f(x)=ax^{p^k}+b$ follows also in this case. When $q$ is odd, and the index of $M$ is even, then a result of Ball, Blokhuis, Brouwer, Storme and SzH onyi combined with a result of McGuire and G"olou{g}lu proves the conjecture. Assume $deg fgeq 1$. We prove that if the size of $D^{-1}D={d^{-1}d' : din Dsetminus {0},, d'in D}$ is less than $q-deg f+2$, then $f$ is a permutation of $mathbb{F}_q$. We use this result to verify the conjecture of Sziklai.
{"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}
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