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Zeros transfer for recursively defined polynomials 递归定义多项式的零转移
Research in Number Theory Pub Date : 2023-11-08 DOI: 10.1007/s40993-023-00480-8
Bernhard Heim, Markus Neuhauser, Robert Tröger
{"title":"Zeros transfer for recursively defined polynomials","authors":"Bernhard Heim, Markus Neuhauser, Robert Tröger","doi":"10.1007/s40993-023-00480-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00480-8","url":null,"abstract":"Abstract The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135342125","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: 1
On further application of the zeta distribution to number theory 论zeta分布在数论中的进一步应用
Research in Number Theory Pub Date : 2023-11-08 DOI: 10.1007/s40993-023-00485-3
Takahiko Fujita, Naohiro Yoshida
{"title":"On further application of the zeta distribution to number theory","authors":"Takahiko Fujita, Naohiro Yoshida","doi":"10.1007/s40993-023-00485-3","DOIUrl":"https://doi.org/10.1007/s40993-023-00485-3","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"27 40","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135391919","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 powerful integers expressible as sums of two coprime fourth powers 在可表示为两个素数四次方和的强整数上
Research in Number Theory Pub Date : 2023-11-07 DOI: 10.1007/s40993-022-00415-9
Noam D. Elkies, Gaurav Goel
{"title":"On powerful integers expressible as sums of two coprime fourth powers","authors":"Noam D. Elkies, Gaurav Goel","doi":"10.1007/s40993-022-00415-9","DOIUrl":"https://doi.org/10.1007/s40993-022-00415-9","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"52 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135474994","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
Spherical designs and modular forms of the $$D_4$$ lattice 球形设计和$$D_4$$晶格的模块化形式
Research in Number Theory Pub Date : 2023-11-01 DOI: 10.1007/s40993-023-00479-1
Masatake Hirao, Hiroshi Nozaki, Koji Tasaka
{"title":"Spherical designs and modular forms of the $$D_4$$ lattice","authors":"Masatake Hirao, Hiroshi Nozaki, Koji Tasaka","doi":"10.1007/s40993-023-00479-1","DOIUrl":"https://doi.org/10.1007/s40993-023-00479-1","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325739","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: 1
Counting elliptic curves over the rationals with a 7-isogeny 计算具有7等构的有理数上的椭圆曲线
Research in Number Theory Pub Date : 2023-10-31 DOI: 10.1007/s40993-023-00482-6
Grant Molnar, John Voight
{"title":"Counting elliptic curves over the rationals with a 7-isogeny","authors":"Grant Molnar, John Voight","doi":"10.1007/s40993-023-00482-6","DOIUrl":"https://doi.org/10.1007/s40993-023-00482-6","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"418 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135869816","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
Computing minimal Weierstrass equations of hyperelliptic curves 超椭圆曲线的最小Weierstrass方程的计算
Research in Number Theory Pub Date : 2023-10-31 DOI: 10.1007/s40993-023-00483-5
Qing Liu
{"title":"Computing minimal Weierstrass equations of hyperelliptic curves","authors":"Qing Liu","doi":"10.1007/s40993-023-00483-5","DOIUrl":"https://doi.org/10.1007/s40993-023-00483-5","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135808249","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: 2
Structure of 2-class groups in the $${mathbb {Z}}_{2}$$-extensions of certain real quadratic fields 某些实数二次域$${mathbb {Z}}_{2}$$ -扩展中2类群的结构
Research in Number Theory Pub Date : 2023-10-21 DOI: 10.1007/s40993-023-00478-2
Jaitra Chattopadhyay, H.  Laxmi, Anupam Saikia
{"title":"Structure of 2-class groups in the $${mathbb {Z}}_{2}$$-extensions of certain real quadratic fields","authors":"Jaitra Chattopadhyay, H.  Laxmi, Anupam Saikia","doi":"10.1007/s40993-023-00478-2","DOIUrl":"https://doi.org/10.1007/s40993-023-00478-2","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"103 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511505","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
Additive arithmetic functions meet the inclusion-exclusion principle, II 加性算术函数满足含不相容原则
Research in Number Theory Pub Date : 2023-10-21 DOI: 10.1007/s40993-023-00477-3
Olivier Bordellès, László Tóth
{"title":"Additive arithmetic functions meet the inclusion-exclusion principle, II","authors":"Olivier Bordellès, László Tóth","doi":"10.1007/s40993-023-00477-3","DOIUrl":"https://doi.org/10.1007/s40993-023-00477-3","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"100 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135512512","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: 3
An explicit upper bound for $$L(1,chi )$$ when $$chi $$ is quadratic 当$$chi $$是二次元时，$$L(1,chi )$$的显式上界
Research in Number Theory Pub Date : 2023-10-03 DOI: 10.1007/s40993-023-00476-4
D. R. Johnston, O. Ramaré, T. Trudgian
{"title":"An explicit upper bound for $$L(1,chi )$$ when $$chi $$ is quadratic","authors":"D. R. Johnston, O. Ramaré, T. Trudgian","doi":"10.1007/s40993-023-00476-4","DOIUrl":"https://doi.org/10.1007/s40993-023-00476-4","url":null,"abstract":"Abstract We consider Dirichlet L -functions $$L(s, chi )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> where $$chi $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>χ</mml:mi> </mml:math> is a non-principal quadratic character to the modulus q . We make explicit a result due to Pintz and Stephens by showing that $$|L(1, chi )|leqslant frac{1}{2}log q$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>⩽</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>log</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> for all $$qgeqslant 2cdot 10^{23}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>2</mml:mn> <mml:mo>·</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>23</mml:mn> </mml:msup> </mml:mrow> </mml:math> and $$|L(1, chi )|leqslant frac{9}{20}log q$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>⩽</mml:mo> <mml:mfrac> <mml:mn>9</mml:mn> <mml:mn>20</mml:mn> </mml:mfrac> <mml:mo>log</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> for all $$qgeqslant 5cdot 10^{50}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>5</mml:mn> <mml:mo>·</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>50</mml:mn> </mml:msup> </mml:mrow> </mml:math> .","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"201 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135739322","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
Arithmetic Dijkgraaf–Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas 算术Dijkgraaf-Witten不变量的实二次域，二次剩余图，和密度公式
Research in Number Theory Pub Date : 2023-09-29 DOI: 10.1007/s40993-023-00471-9
Yuqi Deng, Riku Kurimaru, Toshiki Matsusaka
{"title":"Arithmetic Dijkgraaf–Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas","authors":"Yuqi Deng, Riku Kurimaru, Toshiki Matsusaka","doi":"10.1007/s40993-023-00471-9","DOIUrl":"https://doi.org/10.1007/s40993-023-00471-9","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135194021","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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