Some problems in analytic and computational number theory

点击翻译按钮获取中文摘要

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.