Square-root cancellation for sums of factorization functions over squarefree progressions in $\mathbb F_q[t]$

Abstract

We prove estimates for the level of distribution of the Möbius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to 1 as q goes to ∞, and the power savings converges to square-root cancellation as q goes to ∞. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of L-functions, and the von Mangoldt estimate to one-level densities.

Exposé Bourbaki 1193 : Binary additive problems for polynomials over finite fields (after W. Sawin and M. Shusterman)

“…We conclude with a quick discussion of the results of Sawin [33] concerning the level of distribution of arithmetic functions of polynomials over finite fields. These apply to an extensive class of functions, namely the so-called factorization functions, which are roughly those functions f of polynomials n ∈ k[T] which can be expressed in terms of the factorization pattern of n, i.e., the number of irreducible factors of each degree.…”

“…Again, there is a more precise version (where, for |k| large enough, the constant 3/4 may be replaced by any number < 1) in [33,Th. 1.2], but note the absolute bound for the size of k, independent of k 0 , which highlights a difference with the previous results.…”

“…Example 8.2. -The following table indicates which representations give rise to some of the standard arithmetic functions (in the last line, we have a virtual representation, i.e., we take the corresponding linear combinations for the representations indicated): The main result of Sawin is an explicit upper bound for sums of functions of type f ϱ on arithmetic progressions to squarefree moduli, which translates to level of distribution > 1/2 for individual progressions when |k| is large enough, at least for certain representations ϱ, including those in the table above (see [33,Th. 1.7]):…”

“…We will report in this survey on recent groundbreaking work of W. Sawin and M. Shusterman [37,38] where, for the first time, problems of this kind are solved in the case of polynomials over a fixed finite field, in a very strong quantitative form. Furthermore, we will present results of Sawin [33] where similar ideas are used to prove extremely strong results concerning the level of distribution of arithmetic functions in arithmetic progressions, again in the case of polynomials over fixed finite fields.…”

An induction principle for the Bombieri-Vinogradov theorem over $\mathbb{F}_q[t]$ and a variant of the Titchmarsh divisor problem

“…For instance, our main result allows us to prove equidistribution in arithmetic progressions for almost primes as well as the k-fold divisor function. Pushing the level of distribution beyond half remains an important question in the function field setting as well as demonstrated by recent work due to Sawin [37], [38, Theorem 1.2], as well as Sawin and Shusterman [39,Theorem 1.7].…”