On the distribution of inverses modulo \(n\) (Q1352644)

Let \(n>2\) be an integer. For any integer \(0<a<n\) with \((a,n)=1\) there exists exactly one \(\bar{a}\) with \(0< \bar{a} <n\) such that \(a \bar{a} \equiv 1 \bmod n\). The main purpose of the present paper is to study the asymptotic behaviour of the function \[ S(n,\delta) \colon = |\{ 1 \leq a \leq n-1; (a,n)=1, |a- \bar{a}|< \delta n \} |,\quad 0 < \delta \leq 1\, . \] Using estimates for Kloosterman sums and trigonometric sums the author derives the asymptotic formula \[ S(n,\delta) = \delta (2-\delta) \varphi(n) + O(n^{1/2} \tau^2(n) \log^3 n), \] where \(\varphi\) is Euler's function and \(\tau(n)\) denotes the number of divisors of \(n\).