Abstract:
The number $(q^{n-1}-1)/p$ from $\mathbb{Z}/p\mathbb{Z}$ is called as Fermat quotient (here $p$ is prime and $n$ is nonzero integer). The problems of distribution of Fermat quotients are connected with the upper bounds for
so-called Heilbronn's exponential sum.
The first non-trivial bound for such sum was obtained by Heath-Brown and then
improved by Heath -Brown and Konyagin. The talk is devoted to the recent improvements of
Heath-Brown and Konyagin's result.