Quantitative asymptotics for polynomial patterns in the primes

We prove quantitative estimates for averages of the von Mangoldt and Möbius functions along polynomial progressions $n+P_1(m),\dots ,n+P_k(m)$ for a large class of polynomials $P_i$. 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. 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.