## Abstract

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals $(x,x+{x}^{\u03f5}]$ is about ${x}^{\u03f5}/logx$. The second says that the number of primes $p<x$ in the arithmetic progression $p\equiv a\phantom{\rule{0.1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.1em}{0ex}}d\right)$, for $d<{x}^{1-\delta}$, is about $\frac{\pi \left(x\right)}{\varphi \left(d\right)}$, where $\varphi $ is the Euler totient function.

More precisely, for short intervals we prove: Let $k$ be a fixed integer. Then

$${\pi}_{q}\left(I\right(f,\u03f5\left)\right)\sim \frac{\#I(f,\u03f5)}{k},\phantom{\rule{1.00em}{0ex}}q\to \infty $$ holds uniformly for all prime powers $q$, degree $k$ monic polynomials $f\in {\mathbb{F}}_{q}\left[t\right]$ and ${\u03f5}_{0}(f,q)\le \u03f5$, where ${\u03f5}_{0}$ is either $\frac{1}{k}$, or $\frac{2}{k}$ if $p\mid k(k-1)$, or $\frac{3}{k}$ if further $p=2$ and $\mathrm{deg}\phantom{\rule{0.1em}{0ex}}f\text{'}\le 1$. Here $I(f,\u03f5)=\{g\in {\mathbb{F}}_{q}[t]\mid \mathrm{deg}(f-g)\le \u03f5\mathrm{deg}\phantom{\rule{0.1em}{0ex}}f\}$, and ${\pi}_{q}\left(I\right(f,\u03f5\left)\right)$ denotes the number of prime polynomials in $I(f,\u03f5)$. We show that this estimation fails in the neglected cases.

For arithmetic progressions we prove: let $k$ be a fixed integer. Then

$${\pi}_{q}(k;D,f)\sim \frac{{\pi}_{q}\left(k\right)}{\varphi \left(D\right)},\phantom{\rule{1.00em}{0ex}}q\to \infty ,$$ holds uniformly for all relatively prime polynomials $D,f\in {\mathbb{F}}_{q}\left[t\right]$ satisfying $\Vert D\Vert \le {q}^{k(1-{\delta}_{0})}$, where ${\delta}_{0}$ is either $\frac{3}{k}$ or $\frac{4}{k}$ if $p=2$ and $(f/D)\text{'}$ is a constant. Here ${\pi}_{q}\left(k\right)$ is the number of degree $k$ prime polynomials and ${\pi}_{q}(k;D,f)$ is the number of such polynomials in the arithmetic progression $P\equiv f\phantom{\rule{0.1em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.1em}{0ex}}d\right)$.

We also generalize these results to arbitrary factorization types.

## Citation

Efrat Bank. Lior Bary-Soroker. Lior Rosenzweig. "Prime polynomials in short intervals and in arithmetic progressions." Duke Math. J. 164 (2) 277 - 295, 1 February 2015. https://doi.org/10.1215/00127094-2856728