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Asymptotic For Primitive Roots Producing Polynomials

Let $x \geq 1$ be a large number, let $f(x) \in \mathbb{Z}[x]$ be a prime polynomial of degree $\text{deg}(f)=m$, and let $u\ne \pm 1, v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes $p=f(n) \leq x$ with a fixed primitve root $u$ is derived in this note.
arxiv.org

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Small Prime Primitive Roots in Arithmetic Progressions

Let $p>1$ be a large prime number, let $q=O(\log\log p)$ and let $1\leq a
arxiv.org

Overconvergent unit-root F-isocrystals and isotriviality

We show that a semisimple overconvergent "absolutely unit-root" F-isocrystal on a geometrically connected smooth variety over a finite field becomes constant over a finite covering.
arxiv.org