Prime number theorem for analytic skew products

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus T². More precisely, for every irrational α and every $1$-periodic real analytic g:R of zero mean, let Tα,g : T²² be defined by (x,y) ↦ (x+α,y+g(x)). We prove that if Tα,g is uniquely ergodic then, for every (x,y) ∈ T², the sequence α,gᵖ(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.