Pub Date : 2023-11-02DOI: 10.1142/s1793042124500246
Zachary Feng
We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing [Formula: see text]. Applied to imaginary quadratic fields, this proportion is at least [Formula: see text]. Applied to cyclotomic fields [Formula: see text] with [Formula: see text], this proportion is at least [Formula: see text] with only finitely many exceptions of [Formula: see text], for any choice of [Formula: see text].
{"title":"A Lower Bound on the Proportion of Modular Elliptic Curves Over Galois CM Fields","authors":"Zachary Feng","doi":"10.1142/s1793042124500246","DOIUrl":"https://doi.org/10.1142/s1793042124500246","url":null,"abstract":"We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing [Formula: see text]. Applied to imaginary quadratic fields, this proportion is at least [Formula: see text]. Applied to cyclotomic fields [Formula: see text] with [Formula: see text], this proportion is at least [Formula: see text] with only finitely many exceptions of [Formula: see text], for any choice of [Formula: see text].","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"57 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135875328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1142/s1793042123990014
{"title":"Author index (Volume 19)","authors":"","doi":"10.1142/s1793042123990014","DOIUrl":"https://doi.org/10.1142/s1793042123990014","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139302083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500209
Alex Capunay
{"title":"Computing Shintani Domains","authors":"Alex Capunay","doi":"10.1142/s1793042124500209","DOIUrl":"https://doi.org/10.1142/s1793042124500209","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"127 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500180
Dazhao Tang
{"title":"Congruence properties modulo powers of 2 for overpartitions and overpartition pairs","authors":"Dazhao Tang","doi":"10.1142/s1793042124500180","DOIUrl":"https://doi.org/10.1142/s1793042124500180","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s179304212450043x
Kui Liu, Jie Wu, Zhishan Yang
Denote by $tau$ k (n), $omega$(n) and $mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $omega$, 2 $omega$ , $mu$ 2 , $tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $epsilon$ (x $theta$ f +$epsilon$) for x $rightarrow$ $infty$, where $theta$ $omega$ = 53 110 , $theta$ 2 $omega$ = 9 19 , $theta$ $mu$2 = 2 5 , $theta$ $tau$ k = 5k--1 10k--1 and $epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{`e}s.
表示为 $tau$ K (n)， $omega$(n)及 $mu$ 2 (n) n作为k个自然数乘积的表示形式的个数，n的不同质因数的个数，以及无平方整数的特征函数。设[t]为实数t的积分部分，令f = $omega$， 2 $omega$ ， $mu$ 2、 $tau$ k，我们证明了n x f x n = x d1 f (d) d(d + 1) + 0 $epsilon$ (x) $theta$ F +$epsilon$) for x $rightarrow$ $infty$，其中 $theta$ $omega$ = 53 110， $theta$ 2 $omega$ = 9 19， $theta$ $mu$2 = 2 5， $theta$ $tau$ K = 5k- 1 10k- 1和 $epsilon$ > 0是一个任意小的正数。这些改进了Bordell的相应结果{è}5 .答案:
{"title":"On Some Sums Involving the Integral Part Function","authors":"Kui Liu, Jie Wu, Zhishan Yang","doi":"10.1142/s179304212450043x","DOIUrl":"https://doi.org/10.1142/s179304212450043x","url":null,"abstract":"Denote by $tau$ k (n), $omega$(n) and $mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $omega$, 2 $omega$ , $mu$ 2 , $tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $epsilon$ (x $theta$ f +$epsilon$) for x $rightarrow$ $infty$, where $theta$ $omega$ = 53 110 , $theta$ 2 $omega$ = 9 19 , $theta$ $mu$2 = 2 5 , $theta$ $tau$ k = 5k--1 10k--1 and $epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{`e}s.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135805960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500362
Ertan Elma
{"title":"A Dirichlet Series Related to the Error Term in the Prime Number Theorem","authors":"Ertan Elma","doi":"10.1142/s1793042124500362","DOIUrl":"https://doi.org/10.1142/s1793042124500362","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500210
Xin Wang, Tengyou Zhu
Let $pi$ be a $SL(3,mathbb Z)$ Hecke-Maass cusp form and $chi$ a primitive Dirichlet character of prime power conductor $mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ Lleft(frac{1}{2}+it,pitimes chiright)ll_{pi,varepsilon} p^{3/4}big(mathfrak{q}(1+|t|)big)^{3/4-3/40+varepsilon}, $$ for any $varepsilon>0$ and $t in mathbb{R}$.
{"title":"Hybrid Subconvexity Bounds for Twists of GL(3) <i>L</i>-Functions","authors":"Xin Wang, Tengyou Zhu","doi":"10.1142/s1793042124500210","DOIUrl":"https://doi.org/10.1142/s1793042124500210","url":null,"abstract":"Let $pi$ be a $SL(3,mathbb Z)$ Hecke-Maass cusp form and $chi$ a primitive Dirichlet character of prime power conductor $mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ Lleft(frac{1}{2}+it,pitimes chiright)ll_{pi,varepsilon} p^{3/4}big(mathfrak{q}(1+|t|)big)^{3/4-3/40+varepsilon}, $$ for any $varepsilon>0$ and $t in mathbb{R}$.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135917870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500416
Mahesh Kumar Ram
{"title":"Determination of all Imaginary Cyclic Quartic Fields of Prime Class Number p ≡ 3(mod4), and non-divisibility of class numbers","authors":"Mahesh Kumar Ram","doi":"10.1142/s1793042124500416","DOIUrl":"https://doi.org/10.1142/s1793042124500416","url":null,"abstract":"","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s179304212450026x
Aritram Dhar
A BSTRACT . In this paper, we provide proofs of two 5 ψ 5 summation formulas of Bailey using a 5 φ 4 identity of Carlitz. We show that in the limiting case, the two 5 ψ 5 identities give rise to two 3 ψ 3 summation formulas of Bailey. Finally, we prove the two 3 ψ 3 identities using a technique initially used by Ismail to prove Ramanujan’s 1 ψ 1 summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised 6 ψ 6 sum.
{"title":"On 5ψ5 Identities of Bailey","authors":"Aritram Dhar","doi":"10.1142/s179304212450026x","DOIUrl":"https://doi.org/10.1142/s179304212450026x","url":null,"abstract":"A BSTRACT . In this paper, we provide proofs of two 5 ψ 5 summation formulas of Bailey using a 5 φ 4 identity of Carlitz. We show that in the limiting case, the two 5 ψ 5 identities give rise to two 3 ψ 3 summation formulas of Bailey. Finally, we prove the two 3 ψ 3 identities using a technique initially used by Ismail to prove Ramanujan’s 1 ψ 1 summation formula and later by Ismail and Askey to prove Bailey’s very-well-poised 6 ψ 6 sum.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1142/s1793042124500295
Pramath Anamby
We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_mu$ with either $(mu,2m_1m_2)=1$ or $(mu,2m_1m_2)nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.
{"title":"Non-Vanishing of theta Components of Jacobi Forms with Level and an Application","authors":"Pramath Anamby","doi":"10.1142/s1793042124500295","DOIUrl":"https://doi.org/10.1142/s1793042124500295","url":null,"abstract":"We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_mu$ with either $(mu,2m_1m_2)=1$ or $(mu,2m_1m_2)nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.","PeriodicalId":14293,"journal":{"name":"International Journal of Number Theory","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135918622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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