Quantitative aspects of number theory

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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.