Some problems in analytic and computational number theory

该提案考虑了数论中的两类问题。第一类问题是研究Brun-Titchmarsh定理的一个推广。西格尔-沃尔菲斯证明了他们的著名定理：当gcd(L，q)=1和q&lt；(Logx)^A时，算术进程(AP)中直到x的素数在所有L中均匀分布。在通常的应用中，q与x的比较范围如此短会产生问题。Brun和Titchmarsh证明了在整个Q&lt；x的范围内，Ap中素数to x的个数超过素数到x的个数的1/Q。在这个项目中，我们考虑用至多含有S素数因子的数代替素数来推广Brun-Tichmarsh定理。最近，Chan、曾荫权和这一建议得到了一个类似的Brun-Titichmarsh数论定理，其中q&lt；x和0&lt；S&lt；loglog(x/q)上至多含有S素因子。我们还得到了S的这个值域的一些可能的最好结果，目的之一是把我们的结果改进到S的其他值域和更一般的数，而不是素数或具有固定素数因子的数。

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.