Summary: Let \(p\geq 3\) be a prime. For each primitive root \(x\) modulo \(p\) with \(1\leq x\leq p-1\), it is clear that there exists one and only one primitive root \(\bar x\) modulo \(p\) with \(1\leq\bar x \leq p-1\) such that \(x\bar x\equiv 1 \bmod p\). Let \(\sigma\) be a fixed positive number with \(0\leq \sigma \leq 1\), \(\mathcal A\) denotes the set of all primitive roots modulo \(p\) in interval \([1,p]\). The main purpose of this paper is to study the asymptotic properties of the mean value \[ M(p,k,\sigma)=\mathop{\sum \sum}_{\substack{ a\in\mathcal A\;b\in\mathcal B\\ ab\equiv 1(p)\\ |a-b|<\sigma p}} |a-b|^k \] and give an interesting asymptotic formula.