Some Mean Values Related to Average Multiplicative Orders of Elements in Finite Fields
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Abstract
For any positive integer n let α(n) denote the average order of elements in the cyclic group Z n . In this note, we investigate the functions α(n)/n and α(n)/φ(n) when n ranges through numbers of the form p−1 with p prime, and when n ranges through numbers of the form 2 m −1 with m a positive integer. In particular, we show that such functions have limiting distributions, and we compute their average values, and their minimal and maximal orders.
Keywords
average order enumeration problems cyclic groups finite fields asymptotic estimates Euler’s phi functionPreview
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