Quantitative asymptotics for polynomial patterns in the primes

We prove quantitative estimates for averages of the von Mangoldt and M\"obius functions along polynomial progressions n+P₁(m),…, n+Pₖ(m) for a large class of polynomials Pᵢ. The error terms obtained save an arbitrary power of logarithm, matching the classical Siegel--Walfisz error term. These results give the first quantitative bounds for the Tao--Ziegler polynomial patterns in the primes result, and in the M\"obius case they are new even qualitatively for some collections of polynomials. The proofs are based on a quantitative generalised von Neumann theorem of Peluse, a recent result of Leng on strong bounds for the Gowers uniformity of the primes, and analysis of a ``Siegel model'' for the von Mangoldt function along polynomial progressions.