Generalizations of Artin\u27s conjecture for primitive roots

The dissertation consists of four chapters. The first chapter is an introduction, where we formulate new generalizations of Artin\u27s conjecture for primitive roots and also mention some existing formulations. In chapter two we prove the following result: given a CM elliptic curve E defined over a number field, we have an asymptotic formula counting the number of prime ideals [special characters omitted] with norm ≤ x, such that the reduction of E modulo [special characters omitted] is cyclic. In this formula the error term estimate is given with effectively computable constant. In chapter three we point out that the arguments given in Bilharz\u27s paper for an example of nonexistence of natural density don\u27t seem to be clear. We correct a typo and give full justification for his claim in the cases when the characteristic p of the function field satisfies some inequality. Depending on the interests of the readers, chapters two and three can be read independently. In chapter four we include some examples, which are either in the new formulation or more complicated cases in the existing formulation