Research in Number Theory最新文献

{"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}

{"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}

{"title":"Asymptotic Fermat for signatures (r, r, p) using the modular approach","authors":"Diana Mocanu","doi":"10.1007/s40993-023-00474-6","DOIUrl":"https://doi.org/10.1007/s40993-023-00474-6","url":null,"abstract":"Abstract Let K be a totally real field, and $$rge 5$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;≥&lt;/mml:mo&gt; &lt;mml:mn&gt;5&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; a fixed rational prime. In this paper, we use the modular method as presented in the work of Freitas and Siksek to study non-trivial, primitive solutions $$(x,y,z) in mathcal {O}_K^3$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;x&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;y&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;z&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;∈&lt;/mml:mo&gt; &lt;mml:msubsup&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mi&gt;K&lt;/mml:mi&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:msubsup&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; of the signature ( r , r , p ) equation $$x^r+y^r=z^p$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;x&lt;/mml:mi&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;y&lt;/mml:mi&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;z&lt;/mml:mi&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; (where p is a prime that varies). An adaptation of the modular method is needed, and we follow the work of Freitas which constructs Frey curves over totally real subfields of $$K(zeta _r)$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;K&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;ζ&lt;/mml:mi&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; . When $$K=mathbb {Q}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;K&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mi&gt;Q&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; we get that for most of the primes $$r&lt;150$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;&lt;&lt;/mml:mo&gt; &lt;mml:mn&gt;150&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; with $$r not equiv 1 mod 8$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;r&lt;/mml:mi&gt; &lt;mml:mo&gt;≢&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;mml:mspace /&gt; &lt;mml:mo&gt;mod&lt;/mml:mo&gt; &lt;mml:mspace /&gt; &lt;mml:mn&gt;8&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; there are no non-trivial, primitive integer solutions ( x , y , z ) with 2| z for signatures ( r , r , p ) when p is sufficiently large. Similar results hold for quadratic fields, for example when $$K=mathbb {Q}(sqrt{2})$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;K&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mi&gt;Q&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:msqrt&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:msqrt&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; there are no non-trivial, primitive solutions $$(x,y,z)in mathcal {O}_K^3$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;x&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;y&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;z&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;∈&lt;/mml:mo&gt; &lt;mml:msubsup&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mi&gt;K&lt;/mml:mi&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:msubsup&gt; &lt;/mml","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"90 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":"135193836","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}

{"title":"Explicit upper bounds on the average of Euler–Kronecker constants of narrow ray class fields","authors":"Neelam Kandhil, Rashi Lunia, Jyothsnaa Sivaraman","doi":"10.1007/s40993-023-00472-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00472-8","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387553","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}

{"title":"On equivariant class formulas for Anderson modules","authors":"Tiphaine Beaumont","doi":"10.1007/s40993-023-00473-7","DOIUrl":"https://doi.org/10.1007/s40993-023-00473-7","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134957957","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}

{"title":"Class numbers of certain families of totally real biquadratic fields and a result of Mollin","authors":"Nimish Kumar Mahapatra","doi":"10.1007/s40993-023-00475-5","DOIUrl":"https://doi.org/10.1007/s40993-023-00475-5","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136136081","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}

{"title":"Convergence conditions for p-adic continued fractions","authors":"N. Murru, G. Romeo, Giordano Santilli","doi":"10.1007/s40993-023-00470-w","DOIUrl":"https://doi.org/10.1007/s40993-023-00470-w","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81659236","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}

{"title":"Algebraic and arithmetical properties of Mahler infinite products generated by the second degree polynomials","authors":"D. Duverney, T. Kurosawa","doi":"10.1007/s40993-023-00469-3","DOIUrl":"https://doi.org/10.1007/s40993-023-00469-3","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84302574","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}

{"title":"Sharper bounds for the error term in the prime number theorem","authors":"Andrew Fiori, H. Kadiri, Joshua Swidinsky","doi":"10.1007/s40993-023-00454-w","DOIUrl":"https://doi.org/10.1007/s40993-023-00454-w","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"47 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88661793","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}

{"title":"Reciprocal polynomials and curves with many points over a finite field","authors":"Rohit Gupta, E. A. Mendoza, Luciane Quoos","doi":"10.1007/s40993-023-00464-8","DOIUrl":"https://doi.org/10.1007/s40993-023-00464-8","url":null,"abstract":"","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"36 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86354547","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}