Some problems in analytic and computational number theory

解析数论和计算数论中的一些问题

• 批准号: 238850-2010
• 负责人: Choi, StephenKwokKwong
• 金额: $ 1.09万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2011
• 资助国家: 加拿大
• 起止时间: 2011-01-01 至 2012-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=478994
• 关键词: Some; problems; analytic; computational; number

The proposal considers two types of problems in number theory. The first type of problems is to study an extension to the Brun-Titchmarsh Theorem. Siegel-Walfisz proved their famous theorem that primes up to x in the arithmetic progress (AP) distribute uniformly among all l when gcd(l,q)=1 and q < (log x)^A. In usual application, such a short range of q compared to x creates problems. Brun and Titchmarsh prove that the number of primes to x in the AP is bounded above by 1/q of the numbers of primes up to x, for the whole range of q < x. In this project, we consider an extension to Brun-Tichmarsh Theorem by replacing the primes numbers by the numbers with at most s prime factors. Recently, Chan, Tsang and the proposal were able to obtain an analogous Brun-Titichmarsh Theorem for numbers with at most s prime factors for the range q < x and 0 < s < loglog(x/q). We also obtain some best possible result for this range of s. One aim of this proposal is to improve our result to the other ranges of s and to more general numbers instead of prime numbers or numbers with fixed number of prime factors.

该提案考虑了数论中的两类问题。第一类问题是研究Brun-Titchmarsh定理的扩展。Siegel-Walfisz证明了他们著名的定理，即当gcd（l，q）=1且q <（log x）^A时，算术进程（AP）中直到x的素数均匀分布在所有l中。在通常的应用中，q与x相比如此短的范围会产生问题。Brun和Titchmarsh证明了AP中到x的素数的个数在q < x的整个范围内都有界于到x的素数的个数的1/q。在这个项目中，我们考虑Brun-Tichmarsh定理的一个推广，用至多s个素因子的数代替素数。最近，Chan，Tsang和该提案能够获得一个类似的Brun-Titichmarsh定理，该定理适用于q < x和0 < s < loglog（x/q）范围内至多有s个素因子的数。我们也得到了一些最好的可能的结果为这个范围的s。这个建议的一个目的是改进我们的结果，以其他范围的s和更一般的数字，而不是素数或与固定数量的素因子的数字。