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An integral representation of Catalan numbers using Malmstén's formula 使用马尔姆斯特恩公式的加泰罗尼亚数积分表示法
arXiv - MATH - Number Theory Pub Date : 2024-08-09 DOI: arxiv-2408.05130
Jean-Christophe Pain
{"title":"An integral representation of Catalan numbers using Malmstén's formula","authors":"Jean-Christophe Pain","doi":"arxiv-2408.05130","DOIUrl":"https://doi.org/arxiv-2408.05130","url":null,"abstract":"In this article, we propose an integral expression of the Catalan numbers,\u0000based on Malmst'en's definite-integral representation of\u0000$lnleft[Gamma(x)right]$, $Gamma$ being the usual Gamma function. The\u0000obtained expression is likely to yield new summations involving Catalan numbers\u0000or central binomial coefficients.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"373 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947646","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
New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$ 关于$(-2)^{Ω(n)}$求和函数猜想的新界限和进展
arXiv - MATH - Number Theory Pub Date : 2024-08-08 DOI: arxiv-2408.04143
Daniel R. Johnston, Nicol Leong, Sebastian Tudzi
{"title":"New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$","authors":"Daniel R. Johnston, Nicol Leong, Sebastian Tudzi","doi":"arxiv-2408.04143","DOIUrl":"https://doi.org/arxiv-2408.04143","url":null,"abstract":"In this article, we study the summatory function begin{equation*} W(x)=sum_{nleq x}(-2)^{Omega(n)}, end{equation*} where $Omega(n)$ is the number of prime factors of $n$,\u0000counting multiplicity. We prove $W(x)=O(x)$, and in particular, that\u0000$|W(x)|<2260x$ for all $xgeq 1$. This result provides new progress towards a\u0000conjecture of Sun, which asks whether $|W(x)|<x$ for all $xgeq 3078$. To\u0000obtain our results, we needed to compute new explicit bounds on the Mertens\u0000function $M(x)$. These may be of independent interest. Moreover, we make further conjectures, supported by computational data, that\u0000pertain to the more general function begin{equation*} W_a(x)=sum_{nleq x}(-a)^{Omega(n)} end{equation*} for any real $a>0$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"181 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947710","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
On the $p$-ranks of class groups of certain Galois extensions 论某些伽罗瓦扩展的类群的 $p$-ranks
arXiv - MATH - Number Theory Pub Date : 2024-08-08 DOI: arxiv-2408.04481
Ufuoma Asarhasa, Rusiru Gambheera, Debanjana Kundu, Enrique Nunez Lon-wo, Arshay Sheth
{"title":"On the $p$-ranks of class groups of certain Galois extensions","authors":"Ufuoma Asarhasa, Rusiru Gambheera, Debanjana Kundu, Enrique Nunez Lon-wo, Arshay Sheth","doi":"arxiv-2408.04481","DOIUrl":"https://doi.org/arxiv-2408.04481","url":null,"abstract":"Let $p$ be an odd prime, let $N$ be a prime with $N equiv 1 pmod{p}$, and\u0000let $zeta_p$ be a primitive $p$-th root of unity. We study the $p$-rank of the\u0000class group of $mathbb{Q}(zeta_p, N^{1/p})$ using Galois cohomological\u0000methods and obtain an exact formula for the $p$-rank in terms of the dimensions\u0000of certain Selmer groups. Using our formula, we provide a numerical criterion\u0000to establish upper and lower bounds for the $p$-rank, analogous to the\u0000numerical criteria provided by F.~Calegari--M.~Emerton and\u0000K.~Schaefer--E.~Stubley for the $p$-ranks of the class group of\u0000$mathbb{Q}(N^{1/p})$. In the case $p=3$, we use Redei matrices to provide a\u0000numerical criterion to exactly calculate the $3$-rank, and also study the\u0000distribution of the $3$-ranks as $N$ varies through primes which are $4,7\u0000pmod{9}$.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969681","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
Modular Symbols over Function Fields of Elliptic Curves 椭圆曲线函数域上的模块符号
arXiv - MATH - Number Theory Pub Date : 2024-08-08 DOI: arxiv-2408.04330
Seong Eun Jung
{"title":"Modular Symbols over Function Fields of Elliptic Curves","authors":"Seong Eun Jung","doi":"arxiv-2408.04330","DOIUrl":"https://doi.org/arxiv-2408.04330","url":null,"abstract":"Let $k:=mathbb{F}_q$ be the finite field of $q$ elements and $E$ an elliptic\u0000curve over $k$ given by the equation $f(x,y) = 0$. Let $k[E]$ the affine\u0000coordinate ring and $k(E)$ the field of fractions. We fix a place $infty$ of\u0000$k(E)$ and let $k_{infty}$ be the completion. The group ${rm{GL}}_2(k[E])$\u0000acts on the symmetric space $mathcal{T}$, the Bruhat-Tits building of\u0000${rm{PGL}}_2(k_{infty})$. In this paper, we use the quotient space to\u0000construct the space of modular symbols over $K(E)$. We prove that this space is\u0000given by an explicit set of generators and their relations.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947709","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
Models of CM elliptic curves with a prescribed $ell$-adic Galois image 具有规定$ell$-adic伽罗瓦象的CM椭圆曲线模型
arXiv - MATH - Number Theory Pub Date : 2024-08-08 DOI: arxiv-2408.04159
Enrique González-Jiménez, Álvaro Lozano-Robledo, Benjamin York
{"title":"Models of CM elliptic curves with a prescribed $ell$-adic Galois image","authors":"Enrique González-Jiménez, Álvaro Lozano-Robledo, Benjamin York","doi":"arxiv-2408.04159","DOIUrl":"https://doi.org/arxiv-2408.04159","url":null,"abstract":"For each prime number $ell$ and for each imaginary quadratic order of class\u0000number one or two, we determine all the possible $ell$-adic Galois\u0000representations that occur for any elliptic curve with complex multiplication\u0000by such an order over its minimal field of definition, and then we determine\u0000all the isomorphism classes of elliptic curves that have a prescribed\u0000$ell$-adic Galois representation.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947714","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
Shifted second moment of the Riemann zeta function and a Fourier type kernel 黎曼zeta函数的移位第二矩和傅里叶型核
arXiv - MATH - Number Theory Pub Date : 2024-08-08 DOI: arxiv-2408.04247
Parikshit Dutta, Debashis Ghoshal, Krishnan Rajkumar
{"title":"Shifted second moment of the Riemann zeta function and a Fourier type kernel","authors":"Parikshit Dutta, Debashis Ghoshal, Krishnan Rajkumar","doi":"arxiv-2408.04247","DOIUrl":"https://doi.org/arxiv-2408.04247","url":null,"abstract":"We compute the second moment of the Riemann zeta function for shifted\u0000arguments over a domain that extends the ones in the literature. We use the\u0000Riemann-Siegel formula for the error term in the approximate functional\u0000equation and take the products of all the terms into account. We also show\u0000that, as a function of imaginary shifts on the critical line, the the second\u0000moment behaves like a Fourier-Cauchy type kernel on a class of functions. This\u0000is reminiscent of orthogonal functions.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947721","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
Mordell--Weil groups over large algebraic extensions of fields of characteristic zero 特征为零的域的大代数扩展上的莫德尔--魏尔群
arXiv - MATH - Number Theory Pub Date : 2024-08-07 DOI: arxiv-2408.03495
Takuya Asayama, Yuichiro Taguchi
{"title":"Mordell--Weil groups over large algebraic extensions of fields of characteristic zero","authors":"Takuya Asayama, Yuichiro Taguchi","doi":"arxiv-2408.03495","DOIUrl":"https://doi.org/arxiv-2408.03495","url":null,"abstract":"We study the structure of the Mordell--Weil groups of semiabelian varieties\u0000over large algebraic extensions of a finitely generated field of characteristic\u0000zero. We consider two types of algebraic extensions in this paper; one is of\u0000extensions obtained by adjoining the coordinates of certain points of various\u0000semiabelian varieties; the other is of extensions obtained as the fixed\u0000subfield in an algebraically closed field by a finite number of automorphisms.\u0000Some of such fields turn out to be new examples of Kummer-faithful fields which\u0000are not sub-$p$-adic. Among them, we find both examples of Kummer-faithful\u0000fields over which the Mordell--Weil group modulo torsion can be free of\u0000infinite rank and not free.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947718","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
Zeros of $L$-functions and large partial sums of Dirichlet coefficients L$函数的零点和德里赫特系数的大偏和
arXiv - MATH - Number Theory Pub Date : 2024-08-07 DOI: arxiv-2408.03938
Bryce Kerr, Oleksiy Klurman, Jesse Thorner
{"title":"Zeros of $L$-functions and large partial sums of Dirichlet coefficients","authors":"Bryce Kerr, Oleksiy Klurman, Jesse Thorner","doi":"arxiv-2408.03938","DOIUrl":"https://doi.org/arxiv-2408.03938","url":null,"abstract":"Let $L(s,pi)=sum_{n=1}^{infty}lambda_{pi}(n)n^{-s}$ be an $L$-function\u0000that satisfies a weak form of the generalized Ramanujan conjecture. We prove\u0000that large partial sums of $lambda_{pi}(n)$ strongly repel the low-lying\u0000zeros of $L(s,pi)$ away from the critical line. Our results extend and\u0000quantitatively improve preceding work of Granville and Soundararajan.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947712","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
Optimal sums of three cubes in $mathbb{F}_q[t]$ $mathbb{F}_q[t]$ 中三个立方体的最优和
arXiv - MATH - Number Theory Pub Date : 2024-08-07 DOI: arxiv-2408.03668
Tim Browning, Jakob Glas, Victor Y. Wang
{"title":"Optimal sums of three cubes in $mathbb{F}_q[t]$","authors":"Tim Browning, Jakob Glas, Victor Y. Wang","doi":"arxiv-2408.03668","DOIUrl":"https://doi.org/arxiv-2408.03668","url":null,"abstract":"We use the circle method to prove that a density 1 of elements in\u0000$mathbb{F}_q[t]$ are representable as a sum of three cubes of essentially\u0000minimal degree from $mathbb{F}_q[t]$, assuming the Ratios Conjecture and that\u0000the characteristic is bigger than 3. Roughly speaking, to do so, we upgrade an\u0000order of magnitude result to a full asymptotic formula that was conjectured by\u0000Hooley in the number field setting.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947717","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
Class numbers and integer points on some Pellian surfaces 一些佩利安曲面上的类数和整数点
arXiv - MATH - Number Theory Pub Date : 2024-08-07 DOI: arxiv-2408.03774
Yijie Diao
{"title":"Class numbers and integer points on some Pellian surfaces","authors":"Yijie Diao","doi":"arxiv-2408.03774","DOIUrl":"https://doi.org/arxiv-2408.03774","url":null,"abstract":"We provide an estimate for the number of nontrivial integer points on the\u0000Pellian surface $t^2 - du^2 = 1$ in a bounded region. We give a lower bound on\u0000the size of fundamental solutions for almost all $d$ in a certain class, based\u0000on a recent conjecture of Browning and Wilsch about integer points on log K3\u0000surfaces. We also obtain an upper bound on the average of class number in this\u0000class, assuming the same conjecture.","PeriodicalId":501064,"journal":{"name":"arXiv - MATH - Number Theory","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947649","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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