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Distribution of difference between inverses of consecutive integers modulo \(p\). (English) Zbl 1083.11063

For \(0<n<p\), where \(p\) is a prime, let \(\bar n\) be defined by \(n\bar n\equiv1\pmod p\) with \(0<\bar n<p\). The author proves the following theorem: Let \(\lambda>0\) and \(p\to\infty\). Then \[ \sum_{1\leq n<p-1}| \bar n-\overline{n+1}| ^\lambda= {2p^{\lambda+1}\over(\lambda+1)(\lambda+2)}+O(p^{\lambda+1/2}\log^3p). \] The proof makes use of Kloosterman sums and suitable bounds for exponential sums. It is also shown that, for \(0<k<p\), the number of \(n<p\) such that \(0<\bar n-\overline{n+1}\leq k\) has the asymptotic value \(k-k^2/(2p)\).

MSC:

11N69 Distribution of integers in special residue classes
11L05 Gauss and Kloosterman sums; generalizations
11L07 Estimates on exponential sums
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