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.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OWTU.10012/3844 |
| Date | January 2008 |
| Creators | Camire, Patrice |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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