For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on $\Delta$ in a manner which is consistent with the Hardy-Littlewood Conjecture. We obtain a saving of $q$ if we consider monic polynomials only and $\Delta$ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in $\Delta$. This allows us to obtain additional saving from equidistribution results for $L$-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the Mobius function.

中文翻译：

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$$\mathbb {F}_q[T]$$Fq[T] 上算术函数的相关性

对于固定多项式$\Delta$，我们研究$\mathbb F_q$ 上$n$ 次的多项式$f$ 的数量，使得$f$ 和$f+\Delta$ 都是不可约的，一个$\mathbb F_q[ T]$-孪生素数问题的模拟。在大 $q$ 限制中，如果我们考虑非单项多项式，我们将获得此计数的低阶项，它以与 Hardy-Littlewood 猜想一致的方式依赖于 $\Delta$。如果我们只考虑单项多项式并且 $\Delta$ 是一个标量，我们会节省 $q$。为此，我们使用问题的对称性来免费获得 $\Delta$ 中的少量平均值。这使我们能够从 $L$ 函数的等分布结果中获得额外的节省。我们在一个组合框架中完成所有这些工作，该框架适用于比不可约指标函数更通用的算术函数，