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.


11R58 Arithmetic theory of algebraic function fields
11T06 Polynomials over finite fields
11B25 Arithmetic progressions
Full Text: DOI arXiv Euclid


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