Fermat quotients: Exponential sums, value set and primitive roots

Abstract

For a prime $p$ and an integer $u$ with $\gcd (u,p)=1$, we define Fermat quotients by the conditions $$q_p(u)\equiv \frac{u^{p-1}-1}{p}(\mathrm{mod}p),0\le q_p(u)\le p-1.$$ D. R. Heath-Brown has given a bound of exponential sums with $N$ consecutive Fermat quotients that is nontrivial for $N\ge p^{1/2+\epsilon }$ for any fixed $\epsilon >0$. We use a recent idea of M. Z. Garaev together with a form of the large sieve inequality due to S. Baier and L. Zhao, to show that on average over $p$ one can obtain a nontrivial estimate for much shorter sums starting with $N\ge p^{\epsilon }$. We also obtain lower bounds on the image size of the first $N$ consecutive Fermat quotients and use it to prove that there is a positive integer $n\le p^{3/4+o(1)}$ such that $q_p(n)$ is a primitive root modulo $p$.

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