Research in Number Theory最新文献_第7页

Summing μ ( n ) : a faster elementary algorithm. 求和μ (n):一个更快的初等算法。

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Research in Number Theory Pub Date : 2023-01-01 DOI: 10.1007/s40993-022-00408-8

Harald Andrés Helfgott, Lola Thompson

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Summing <ns0:math><ns0:mrow><ns0:mi>μ</ns0:mi> <ns0:mo>(</ns0:mo> <ns0:mi>n</ns0:mi> <ns0:mo>)</ns0:mo></ns0:mrow> </ns0:math> : a faster elementary algorithm.","authors":"Harald Andrés Helfgott, Lola Thompson","doi":"10.1007/s40993-022-00408-8","DOIUrl":"https://doi.org/10.1007/s40993-022-00408-8","url":null,"abstract":"We present a new elementary algorithm that takes <math><mrow><mtext>time</mtext> <mspace></mspace> <mspace></mspace> <msub><mi>O</mi> <mi>ϵ</mi></msub> <mfenced><msup><mi>x</mi> <mfrac><mn>3</mn> <mn>5</mn></mfrac> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mrow><mfrac><mn>8</mn> <mn>5</mn></mfrac> <mo>+</mo> <mi>ϵ</mi></mrow> </msup> </mfenced> <mspace></mspace> <mspace></mspace> <mtext>and</mtext> <mspace></mspace> <mtext>space</mtext> <mspace></mspace> <mspace></mspace> <mi>O</mi> <mfenced><msup><mi>x</mi> <mfrac><mn>3</mn> <mn>10</mn></mfrac> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mfrac><mn>13</mn> <mn>10</mn></mfrac> </msup> </mfenced> </mrow> </math> (measured bitwise) for computing <math><mrow><mi>M</mi> <mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> <mo>=</mo> <msub><mo>∑</mo> <mrow><mi>n</mi> <mo>≤</mo> <mi>x</mi></mrow> </msub> <mi>μ</mi> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> <mo>,</mo></mrow> </math> where <math><mrow><mi>μ</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </math> is the Möbius function. This is the first improvement in the exponent of x for an elementary algorithm since 1985. We also show that it is possible to reduce space consumption to <math><mrow><mi>O</mi> <mo>(</mo> <msup><mi>x</mi> <mrow><mn>1</mn> <mo>/</mo> <mn>5</mn></mrow> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mrow><mn>5</mn> <mo>/</mo> <mn>3</mn></mrow> </msup> <mo>)</mo></mrow> </math> by the use of (Helfgott in: Math Comput 89:333-350, 2020), at the cost of letting time rise to the order of <math> <mrow><msup><mi>x</mi> <mrow><mn>3</mn> <mo>/</mo> <mn>5</mn></mrow> </msup> <msup><mrow><mo>(</mo> <mo>log</mo> <mi>x</mi> <mo>)</mo></mrow> <mn>2</mn></msup> <mo>log</mo> <mo>log</mo> <mi>x</mi></mrow> </math> .","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"9 1","pages":"6"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9731940/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10687746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Geometric sieve over number fields for higher moments. 在数字域上进行几何筛选以获得更高的矩。

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Research in Number Theory Pub Date : 2023-01-01 Epub Date: 2023-08-02 DOI: 10.1007/s40993-023-00466-6

Giacomo Micheli, Severin Schraven, Simran Tinani, Violetta Weger

{"title":"Geometric sieve over number fields for higher moments.","authors":"Giacomo Micheli, Severin Schraven, Simran Tinani, Violetta Weger","doi":"10.1007/s40993-023-00466-6","DOIUrl":"10.1007/s40993-023-00466-6","url":null,"abstract":"The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number field K. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":"9 3","pages":"62"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10397131/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9943903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

p-adic vertex operator algebras. p进顶点算子代数。

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Research in Number Theory Pub Date : 2023-01-01 DOI: 10.1007/s40993-023-00433-1

Cameron Franc, Geoffrey Mason

引用次数: 2

Kampé de Fériet hypergeometric functions over finite fields 有限域上的超几何函数

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Research in Number Theory Pub Date : 2022-12-29 DOI: 10.1007/s40993-023-00460-y

Ryojun Ito, Satoshi Kumabe, Akio Nakagawa, Yusuke Nemoto

引用次数: 0

Fast change of level and applications to isogenies 水平的快速变化和应用到同基因

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Research in Number Theory Pub Date : 2022-12-12 DOI: 10.1007/s40993-022-00407-9

D. Lubicz, Damien Robert

引用次数: 12

Triangular modular curves of small genus 小属的三角形模曲线

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Research in Number Theory Pub Date : 2022-11-29 DOI: 10.1007/s40993-022-00395-w

Juanita Duque-Rosero, J. Voight

引用次数: 0

Rogers–Ramanujan type generalized reciprocal identities and Eisenstein series Rogers-Ramanujan型广义互反恒等式与爱森斯坦级数

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Research in Number Theory Pub Date : 2022-11-29 DOI: 10.1007/s40993-022-00416-8

Zafer Selcuk Aygin

引用次数: 0

Modular forms via invariant theory 模形式通过不变理论

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Research in Number Theory Pub Date : 2022-11-10 DOI: 10.1007/s40993-023-00442-0

Fabien Cl'ery, G. van der Geer

引用次数: 1

Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3 3属超椭圆曲线雅可比矩阵扭转子群的计算

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Research in Number Theory Pub Date : 2022-11-07 DOI: 10.1007/s40993-023-00429-x

J. Müller, Berno Reitsma

引用次数: 1

Correction: Examples of abelian surfaces failing the local-global principle for isogenies 更正:阿贝尔曲面不符合局部-全局原理的例子

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Research in Number Theory Pub Date : 2022-10-26 DOI: 10.1007/s40993-022-00406-w

Barinder S. Banwait

引用次数: 0