## Prime polynomials in short intervals and in arithmetic progressions.(English)Zbl 1395.11132

Summary: 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 $$(x,x+x^\varepsilon]$$ is about $$x^\varepsilon/\log x$$. The second says that the number of primes $$p<x$$ in the arithmetic progression $$p\equiv a\pmod d$$, for $$d<x^{1-\delta}$$, is about $$\frac{\pi(x)}{\varphi(d)}$$, where $$\varphi$$ is the Euler totient function.
More precisely, for short intervals we prove: Let $$k$$ be a fixed integer. Then
$\pi_q(I(f,\varepsilon))\sim\frac{\#I(f,\varepsilon)}{k}, \quad q\to\infty$
holds uniformly for all prime powers $$q$$, degree $$k$$ monic polynomials $$f\in \mathbb F_q[t]$$ and $$\varepsilon_0(f,q)\leq\varepsilon$$, where $$\varepsilon_0$$ is either $$\frac{1}{k}$$, or $$\frac{2}{k}$$ if $$p\mid k(k-1)$$, or $$\frac{3}{k}$$ if further $$p=2$$ and deg$$f'\leq1$$. Here $$I(f,\varepsilon)=\{g\in \mathbb F_q[t]\mid \mathrm{deg}(f-g)\le \varepsilon\, \mathrm{deg}f\}$$, and $$\pi_q(I(f,\varepsilon))$$ denotes the number of prime polynomials in $$I(f,\varepsilon)$$. We show that this estimation fails in the neglected cases.
For arithmetic progressions we prove: let $$k$$ be a fixed integer. Then
$\pi_q(k;D,f )\sim \frac{\pi_q(k)}{\varphi(D)},\quad q\to\infty,$
holds uniformly for all relatively prime polynomials $$D,f\in\mathbb F_q[t]$$ satisfying $$\|D\|\le q^{k(1-\delta_0)}$$, where $$\delta_0$$ is either $$\frac3k$$ or $$\frac4k$$ 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 $$P\equiv f \pmod d$$.
We also generalize these results to arbitrary factorization types.

### MSC:

 11R58 Arithmetic theory of algebraic function fields 11T06 Polynomials over finite fields 11B25 Arithmetic progressions

### Keywords:

arithmetic progressions; prime polynomials
Full Text:

### References:

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