Fermat quotients: Exponential sums, value set and primitive roots

For a prime p and an integer u with gcd(u, p)=1, we define Fermat quotients by the conditions Heath-Brown has given a bound of exponential sums with N consecutive Fermat quotients that is non-trivial for N≥p ^1/2+ε for any fixed ε>0. We use a recent idea of Garaev together with a form of the large sieve inequality due to Baier and Zhao to show that, on average over p, one can obtain a non-trivial estimate for much shorter sums starting with N≥p ^ε. We also obtain lower bounds on the image size of the first N consecutive Fermat quotients and use them to prove that there is a positive integer n≤p ^3/4+o(1) such that qp(n) is a primitive root modulo p.