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 $(x,x+x^{ϵ}]$ is about $x^{ϵ}/logx$. The second says that the number of primes $p<x$ in the arithmetic progression $p\equiv a\left(\mathrm{mod}d\right)$, 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

$${\pi }_q\left(I\right(f,ϵ\left)\right)\sim \frac{\#I(f,ϵ)}{k},q\to \infty$$ holds uniformly for all prime powers $q$, degree $k$ monic polynomials $f\in {\mathbb{F}}_q\left[t\right]$ and ${ϵ}_0(f,q)\le ϵ$, where ${ϵ}_0$ is either $\frac{1}{k}$, or $\frac{2}{k}$ if $p\mid k(k-1)$, or $\frac{3}{k}$ if further $p=2$ and $\mathrm{deg}f\text{'}\le 1$. Here $I(f,ϵ)=\{g\in {\mathbb{F}}_q[t]\mid \mathrm{deg}(f-g)\le ϵ\mathrm{deg}f\}$, and ${\pi }_q\left(I\right(f,ϵ\left)\right)$ denotes the number of prime polynomials in $I(f,ϵ)$. We show that this estimation fails in the neglected cases.

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

$${\pi }_q(k;D,f)\sim \frac{{\pi }_q\left(k\right)}{\varphi \left(D\right)},q\to \infty ,$$ 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 $P\equiv f\left(\mathrm{mod}d\right)$.

We also generalize these results to arbitrary factorization types.