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Abstract
This thesis discusses three topics in Number Theory, where the results are obtained by using analytic methods. First, we discuss results on the distribution of the small prime power residues modulo a prime. We give an elementary approach by using reciprocity laws first, and then we discuss how to improve and generalize these results using character sum estimates and sieve results. Secondly, we discuss a quantitative improvement on the number of weakly prime numbers, the primes that change to a composite number after altering a single digit. Finally, we discuss an asymptotic formula for the number of pure fields of fixed prime degree and bounded discriminant.