TY - JOUR

T1 - Prime number theorem for analytic skew products

AU - Kanigowski, Adam

AU - Lemańczyk, Mariusz

AU - Radziwiłł, Maksym

N1 - Publisher Copyright:
© 2024 Department of Mathematics, Princeton University. All Rights Reserved.

PY - 2024

Y1 - 2024

N2 - We establish a prime number theorem for all uniquely ergodic, analytic skew products on the 2-torus T2. More precisely, for every irrational a and every 1-periodic real analytic g: R → R of zero mean, let Tα,g: T2 → T2 be defined by (x,y) → (x + α, y + g(x)). We prove that if Tα,g is uniquely ergodic then, for every (x,y) Є T2, the sequence {Tpa,g(x, y)} is equidistributed on T2 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.

AB - We establish a prime number theorem for all uniquely ergodic, analytic skew products on the 2-torus T2. More precisely, for every irrational a and every 1-periodic real analytic g: R → R of zero mean, let Tα,g: T2 → T2 be defined by (x,y) → (x + α, y + g(x)). We prove that if Tα,g is uniquely ergodic then, for every (x,y) Є T2, the sequence {Tpa,g(x, y)} is equidistributed on T2 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.

KW - multiplicative number theory

KW - prime number theorem

KW - skew product

KW - smooth dynamical systems

UR - http://www.scopus.com/inward/record.url?scp=85188234316&partnerID=8YFLogxK

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U2 - 10.4007/annals.2024.199.2.2

DO - 10.4007/annals.2024.199.2.2

M3 - Article

AN - SCOPUS:85188234316

SN - 0003-486X

VL - 199

SP - 591

EP - 705

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 2

ER -