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 $$q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1.$$ 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 $p$

Periodic Structure of the Exponential Pseudorandom Number Generator

We investigate the periodic structure of the exponential pseudorandom number generator obtained from the map $x\mapsto g^x\pmod p$ that acts on the set $\{1, \ldots, p-1\}$

Congruences with intervals and subgroups modulo a prime

We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014)

Adelic Openness for Drinfeld Modules in Special Characteristic

For any Drinfeld module of special characteristic p0 over a finitely generated field, we study the associated adelic Galois representation at all places different from p0 and \infty, and determine the image of the geometric Galois group up to commensurability