On the Chowla and twin primes conjectures over Fq[T]

• 成果类型: Article
• 署名作者: Sawin, Will; Shusterman, Mark
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2022.196.2.1
• 发表日期: 2022
• 页码: 457-506
• 关键词: multiplicative functions; bounded gaps; POLYNOMIALS; VALUES; number

摘要

Using geometric methods, we improve on the function field version of the Burgess bound and show that, when restricted to certain special subspaces, the Mobius function over F-q[T] can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to 1 for the Mobius function in arithmetic progressions and resolve Chowla's k-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the Mobius function, we obtain a level of distribution beyond 1/2 for irreducible polynomials, and establish the twin prime conjecture in a quantitative form. All these results hold for finite fields satisfying a simple condition.