Prime number theorem for analytic skew products

• 成果类型: Article
• 署名作者: Kanigowski, Adam; Lemanczyk, Mariusz; Radziwill, Maksym
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.199.2.2
• 发表日期: 2024
• 页码: 591-705
• 关键词: mobius function; gaps

摘要

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the 2 -torus T 2 . More precisely, for every irrational alpha and every 1 -periodic real analytic g : R -+ R of zero mean, let T alpha,g : T 2 -+ T 2 be defined by ( x, y ) 7-+ ( x + alpha, y + g ( x )). We prove that if T alpha,g is uniquely ergodic then, for every ( x, y ) is an element of T 2 , the sequence { T p alpha,g ( x, y )} is equidistributed on T 2 as p traverses prime numbers. This is the first example of a class of natural, non -algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if g is only continuous on T .