If we fix an integer a not equal to -1 and which is not a perfect square, we are interested in estimating the quantity N_{a}(x) representing the number of prime integers p up to x such that a is a generator of the cyclic group (Z/pZ)*. We will first show how to obtain an aymptotic formula for N_{a}(x) under the assumption of the generalized Riemann hypothesis. We then investigate the average behaviour of N_{a}(x) and we provide an unconditional result. Finally, we discuss how to generalize the problem over (Z/mZ)*, where m > 0 is not necessarily a prime integer. We present an average result in this setting and prove the existence of oscillation.

If we fix an integer a not equal to -1 and which is not a perfect square, we are interested in estimating the quantity N_{a}(x) representing the number of prime integers p up to x such that a is a generator of the cyclic group (Z/pZ)*. We will first show how to obtain an aymptotic formula for N_{a}(x) under the assumption of the generalized Riemann hypothesis. We then investigate the average behaviour of N_{a}(x) and we provide an unconditional result. Finally, we discuss how to generalize the problem over (Z/mZ)*, where m > 0 is not necessarily a prime integer. We present an average result in this setting and prove the existence of oscillation.