[Paper Review] Bounded gaps between primes in number fields and function fields

Abel Castillo, Chris Hall+3 more|arXiv (Cornell University)|Mar 23, 2014

Analytic Number Theory Research6 references4 citations

TL;DR

This paper extends the Maynard-Tao theorem on bounded gaps between primes to number fields and function fields over finite fields. By adapting the Maynard-Tao method to the number field setting and proving an analogue in the function field $\mathbb{F}_q(t)$, the authors establish the existence of infinitely many bounded gaps between prime elements in rings of integers of number fields and between monic irreducible polynomials in $\mathbb{F}_q[t]$, with explicit bounds depending only on the number of complex embeddings or independent of $q$. The key result is a uniform $k_0(2) = 105$ for function fields, enabling bounded gaps of degree 600 in totally real fields and positive density of gap polynomials.

ABSTRACT

The Hardy--Littlewood prime $k$-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field $\mathbb{F}_q(t)$.

Motivation & Objective

• To extend the Maynard-Tao theorem on bounded gaps between primes to the context of number fields and function fields.
• To establish the existence of infinitely many bounded gaps between prime elements in the ring of integers of a number field $K$, particularly in totally real fields.
• To prove an analogue of the Maynard-Tao theorem in the function field $\mathbb{F}_q(t)$, where irreducible polynomials play the role of primes.
• To derive quantitative bounds on the size of gaps between prime elements in number fields and between irreducible polynomials in function fields.
• To show that the required $k_0$-value for bounded gaps in function fields is independent of $q$, allowing uniform bounds across all finite fields.

Proposed method

• Adapt the Maynard-Tao method to number fields by defining admissibility of $k$-tuples modulo prime ideals $\mathfrak{p}$ in $\mathcal{O}_K$, ensuring the tuple does not cover $\mathcal{O}_K/\mathfrak{p}$.
• Prove that for any $m \geq 2$, there exists $k_0 = k_0(m,K)$ such that any admissible $k$-tuple in $\mathcal{O}_K$ with $k \geq k_0$ yields infinitely many $\alpha \in \mathcal{O}_K$ for which at least $m$ of $\alpha + h_i$ are prime.
• Define admissibility in $\mathbb{F}_q[t]$ by requiring that for each irreducible $P$, the set $\{h_1,\dots,h_k\}$ does not cover $\mathbb{F}_q[t]/P$, and apply the Maynard-Tao method to this setting.
• Show that the required $k_0(m)$ for bounded gaps in $\mathbb{F}_q(t)$ is independent of $q$, so that Maynard's values such as $k_0(2) = 105$ are valid.
• Use the level of distribution of primes in $K$ and the fact that $K$ totally real implies $\theta < 1/2$ to apply Maynard's results on $M_k$-values.
• Establish density results for gap polynomials by induction on $k$, showing that the proportion of degree-$d$ polynomials appearing as gaps in degree-$n$ irreducibles is at least $\frac{1}{k-1} - \frac{1}{q-1}$.

Experimental results

Research questions

• RQ1 Can the Maynard-Tao method for bounded gaps between primes be generalized to number fields, and what conditions ensure such gaps exist in $\mathcal{O}_K$?
• RQ2 What is the minimal $k_0$ required to guarantee bounded gaps between $m$ prime elements in the ring of integers of a number field $K$?
• RQ3 Does the Maynard-Tao method extend to function fields $\mathbb{F}_q(t)$, and can the required $k_0$ be chosen independently of $q$?
• RQ4 What bounds can be established on the size of gaps between monic irreducible polynomials in $\mathbb{F}_q[t]$ of fixed large degree $n$?
• RQ5 What proportion of degree-$d$ polynomials can appear as gaps between monic irreducibles of degree $n$ in $\mathbb{F}_q[t]$?

Key findings

• For any totally real number field $K$, there are infinitely many prime elements $\alpha_1, \alpha_2 \in \mathcal{O}_K$ such that $|\sigma(\alpha_1 - \alpha_2)| \leq 600$ for every embedding $\sigma$ of $K$, based on an admissible 105-tuple of rational integers with diameter 600.
• The value $k_0(2) = 105$ suffices for bounded gaps between irreducible polynomials in $\mathbb{F}_q[t]$, independent of $q$, extending Maynard's result to function fields.
• For any $d$, the proportion of degree-$d$ polynomials that appear as gaps between monic irreducibles of degree $n$ in $\mathbb{F}_q[t]$ is at least $\frac{1}{k_0-1} - \frac{1}{q-1}$ for sufficiently large $n$, with $k_0 = k_0(2)$.
• Every monomial $a \cdot t^d$ occurs as a gap between monic irreducibles of degree $n$ for all sufficiently large $n$ with $(n-d, q-1) = 1$, provided $q \geq k_0 + 1$.
• The existence of bounded gaps in $\mathbb{F}_q(t)$ is guaranteed for any $q$ and $k_0(2) = 105$, with the same $k_0$ as in the classical case.
• The method ensures that for any admissible $k$-tuple in $\mathbb{F}_q[t]$ with $k \geq k_0$, there are infinitely many $f \in \mathbb{F}_q[t]$ such that at least $m$ of $f + h_i$ are irreducible, with $k_0$ independent of $q$.