Quantitative aspects of number theory

这个项目的目的是研究数论中具有定量成分的问题。通常，问题会涉及一些代数对象，如代数数域或代数簇上的有理点。然而，我们提出的问题是分析性质的。例如，有多少个给定类型和给定属性的数域，使得它们的判别式以（大）数B为界？答案是，我们正在寻求然后将是一个渐近公式，或者，如果这是太难了，这个数字的界限，因为B趋于infinites.the具体的问题，研究的学生包括精确的信息，分布的数量分歧素数在阿贝尔数域，以及分布的某些非正常的扩展与规定的规范。这需要巧妙地综合新颖的代数、分析和概率思想。

The aim of this project is to investigate problems in number theory that have a quantitative component. Typically, the problems would involve some algebraic objects, such as as algebraic number fields or rational points on algebraic varieties. The questions that we ask are, however, of analytic nature. For example, how many number fields are there of a given type and with given properties, such that their discriminant is bounded by a (large) number B? The answer that we are seeking would then be an asymptotic formula or, if this is too hard, bounds for this number as B tends to infinity.The concrete questions studied by the student include precise information on the distribution of the number of ramified primes in abelian number fields, as well as the distribution of certain non-normal extensions with prescribed norms. This requires a clever synthesis of novel algebraic, analytic and probabilistic ideas.