## 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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