Some problems in analytic and computational number theory

该建议考虑数论中的两类问题。第一类问题是研究Brun-Titchmarsh定理的推广。Siegel-Walfisz证明了他们的著名定理，即当gcd(l,q)=1且q < (log x)^A时，算术进程（AP）中x以内的质数均匀分布于所有l中。在通常的应用中，与x相比，如此短的q范围会产生问题。Brun和Titchmarsh证明了在整个q < x的范围内，AP中小于x的质数的个数以小于x的质数的1/q为上界。在本课题中，我们考虑将Brun- tichmarsh定理的推广，用最多有s个质因数的数来代替素数。最近，Chan， Tsang和该建议能够对q < x和0 < s < logog （x/q）范围内最多有s个素数因子的数获得一个类似的brown - titichmarsh定理。本文的一个目的是将我们的结果改进到s的其他范围和更一般的数，而不是质数或具有固定数量的质因数的数。

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.