## Duke Mathematical Journal

### Prime polynomials in short intervals and in arithmetic progressions

#### 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 $(x,x+x^{\epsilon}]$ is about $x^{\epsilon}/\log x$. The second says that the number of primes $p\lt x$ in the arithmetic progression $p\equiv a\ (\mathrm{mod}\ d)$, for $d\lt x^{1-\delta}$, is about $\frac{\pi(x)}{\phi(d)}$, where $\phi$ is the Euler totient function.

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

$$\pi_{q}(I(f,\epsilon))\sim\frac{\#I(f,\epsilon)}{k},\quad q\to\infty$$ holds uniformly for all prime powers $q$, degree $k$ monic polynomials $f\in\mathbb{F}_{q}[t]$ and $\epsilon_{0}(f,q)\leq\epsilon$, where $\epsilon_{0}$ is either $\frac{1}{k}$, or $\frac{2}{k}$ if $p\mid k(k-1)$, or $\frac{3}{k}$ if further $p=2$ and $\mathrm{deg}\ f'\leq1$. Here $I(f,\epsilon)=\{g\in\mathbb{F}_{q}[t]\mid\mathrm{deg}\ (f-g)\leq\epsilon\mathrm{deg}\ f\}$, and $\pi_{q}(I(f,\epsilon))$ denotes the number of prime polynomials in $I(f,\epsilon)$. We show that this estimation fails in the neglected cases.

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

$$\pi_{q}(k;D,f)\sim\frac{\pi_{q}(k)}{\phi(D)},\quad q\to\infty,$$ holds uniformly for all relatively prime polynomials $D,f\in\mathbb{F}_{q}[t]$ satisfying $\Vert D\Vert \leq q^{k(1-\delta_{0})}$, where $\delta_{0}$ is either $\frac{3}{k}$ or $\frac{4}{k}$ if $p=2$ and $(f/D)'$ is a constant. Here $\pi_{q}(k)$ is the number of degree $k$ prime polynomials and $\pi_{q}(k;D,f)$ is the number of such polynomials in the arithmetic progression $P\equiv f\ (\mathrm{mod}\ d)$.

We also generalize these results to arbitrary factorization types.

#### Article information

Source
Duke Math. J., Volume 164, Number 2 (2015), 277-295.

Dates
First available in Project Euclid: 30 January 2015

https://projecteuclid.org/euclid.dmj/1422627049

Digital Object Identifier
doi:10.1215/00127094-2856728

Mathematical Reviews number (MathSciNet)
MR3306556

Zentralblatt MATH identifier
06416949

Subjects
Primary: 11T06: Polynomials

#### Citation

Bank, Efrat; Bary-Soroker, Lior; Rosenzweig, Lior. Prime polynomials in short intervals and in arithmetic progressions. Duke Math. J. 164 (2015), no. 2, 277--295. doi:10.1215/00127094-2856728. https://projecteuclid.org/euclid.dmj/1422627049

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