Fermat quotients: Exponential sums, value set and primitive roots

TL;DR

This paper improves bounds on exponential sums involving Fermat quotients, analyzes their value set, and demonstrates the existence of primitive roots among these quotients for certain integers.

Contribution

It introduces new average bounds for shorter sums of Fermat quotients and establishes the existence of primitive roots within their value set.

Findings

Nontrivial bounds for sums with N ≥ p^ε on average over p

Lower bounds on the size of the Fermat quotient value set

Existence of a primitive root among Fermat quotients for some n ≤ p^{3/4 + o(1)}

Abstract

For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ 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 $p$.