Prime and M\"obius correlations for very short intervals in $\mathbb{F}_q[x]$

TL;DR

This paper studies the distribution of primes and M"obius function correlations in very short intervals over function fields, revealing conditions for independence and examples of failure, with implications for analogs of classical number theory conjectures.

Contribution

It establishes new results on prime distribution and M"obius correlations in function fields, including independence conditions and counterexamples, extending classical conjectures to this setting.

Findings

Independence holds for generic Morse polynomials.

Examples show failure of independence in certain cases.

Square root cancellation in M"obius sums is equivalent to that in Chowla sums.

Abstract

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the M\"obius $\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for "generic" set of shifts. We can also exhibit examples for which there is no cancellation at all in M\"obius/Chowla type sums (in fact, it turns out that (square root) cancellation in M\"obius sums is {\em equivalent} to (square root) cancellation in Chowla type sums), as well as intervals where the heuristic "primes are independent" fails badly. The results are deduced from a general theorem on correlations of arithmetic class functions; these include characteristic functions on primes, the M\"obius $\mu$ function, and divisor functions (e.g., function field analogs of the Titchmarsh divisor problem can be treated.) We also prove analogous, but slightly weaker, results in the more delicate fixed characteristic setting, i.e., for $f(x) \in \mathbb{F}_q[x]$ and intervals of the form $f(x) +a$ for $a \in \mathbb{F}_q$, where $p$ is fixed and $q=p^{l}$ grows.