APPLICATIONS OF PRIME DENSITIES IN NUMBER THEORY AND CLASSIFICATION OF NUMBER FIELDS WITH BOUNDED INVARIANTS

This Ph.D. thesis collects the author's works and interests in several parts of Number Theory, from algebraic problems related to relations between number fields which are based on the factorization of prime numbers in the rings of integers, up to the application of tools concerning the density of primes with given splitting type in number fields to the computation of the average rank of specific families of elliptic curves, concluding finally with the classification and estimate of the main invariants of a number field, like the discriminant and the regulator, pursued by means of analytic formulas and algorithmic methods developed on the previous tools and implemented on suitable computer algebra systems, like PARI/GP.