Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions

We establish quantitative bounds on the Uᵏ[N] Gowers norms of the Möbius function μ and the von Mangoldt function Λ for all k , with error terms of the shape O((loglog N)⁻ᶜ) . As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms. We also obtain the first quantitative bounds on the size of sets containing no k -term arithmetic progressions with shifted prime difference.