Pub Date : 2023-11-08DOI: 10.1007/s40993-023-00480-8
Bernhard Heim, Markus Neuhauser, Robert Tröger
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.
D 'Arcais多项式(也称为Nekrasov-Okounkov多项式)的零表示Dedekind函数幂的傅里叶系数的消失。这些多项式满足非常系数遗传型差分方程。我们将D 'Arcais多项式与满足卷积型Volterra差分方程的多项式联系起来。我们得到了关于零位置转移的结果。作为应用，我们得到了第二类切比雪夫多项式与1相关拉盖尔多项式之间的恒等式。我们得到了Lehmer猜想的一个新版本和Hermite多项式的零点界。
{"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}
Pub Date : 2023-11-08DOI: 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}
Pub Date : 2023-11-07DOI: 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}
Pub Date : 2023-11-01DOI: 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}
Pub Date : 2023-10-31DOI: 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}
Pub Date : 2023-10-31DOI: 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}
Pub Date : 2023-10-21DOI: 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}
Pub Date : 2023-10-21DOI: 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}
Pub Date : 2023-10-03DOI: 10.1007/s40993-023-00476-4
D. R. Johnston, O. Ramaré, T. Trudgian
Abstract We consider Dirichlet L -functions $$L(s, chi )$$ L(s,χ) where $$chi $$ χ is a non-principal quadratic character to the modulus q . We make explicit a result due to Pintz and Stephens by showing that $$|L(1, chi )|leqslant frac{1}{2}log q$$ |L(1,χ)|⩽12logq for all $$qgeqslant 2cdot 10^{23}$$ q⩾2·1023 and $$|L(1, chi )|leqslant frac{9}{20}log q$$ |L(1,χ)|⩽920logq for all $$qgeqslant 5cdot 10^{50}$$ q⩾5·1050 .
我们考虑Dirichlet L -函数$$L(s, chi )$$ L (s， χ)，其中$$chi $$ χ是模q的非主二次特征。我们通过显示所有$$qgeqslant 2cdot 10^{23}$$ q大于或等于2·10 23的$$|L(1, chi )|leqslant frac{1}{2}log q$$ | L (1， χ) |≤1 2 log q和所有$$qgeqslant 5cdot 10^{50}$$ q大于或等于5·10 50的$$|L(1, chi )|leqslant frac{9}{20}log q$$ | L (1， χ) |≤9 20 log q来明确pinz和Stephens的结果。
{"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}
Pub Date : 2023-09-29DOI: 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}
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