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On the distribution of primitive roots mod \(p\). (English) Zbl 0892.11003

Let \(p\) be a large prime number and consider the set of primitive roots mod \(p\) with representatives \(1 < \gamma_1 < \cdots < \gamma_{\varphi(p-1)} < p\). The authors show that, if \(\varphi(p-1)/p\) is small, then the distribution of the number of \(\gamma\)’s in the interval \((n,n+t]\), \(1 \leq n \leq p\), where \(t \sim \lambda p/ \varphi(p-1)\), \(\lambda >0\), is approximately that of a Poisson variable with parameter \(\lambda\). Moreover, they prove that, for any sequence \(\{p_n \}_{n \geq 1}\) of primes with \(\varphi(p_n -1)/p_n \to 0\), the sequence \[ \biggl\{ \frac{| \{i;\;1 \leq i \leq \varphi(p_n-1), \gamma_{i+1} - \gamma_i \geq \lambda p_n / \varphi(p_n-1) \} | }{\varphi(p_n-1)} \biggr\}_{n \geq 1} \] of functions converges to \(e^{- \lambda}\) uniformly on compact subsets of \([0, \infty)\).

MSC:

11A07 Congruences; primitive roots; residue systems
11N69 Distribution of integers in special residue classes
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