Quantitative asymptotics for polynomial patterns in the primes

TL;DR

This paper provides the first quantitative bounds for polynomial patterns in primes, improving error estimates to match classical results and advancing understanding of prime distributions along polynomial sequences.

Contribution

It introduces new quantitative bounds for polynomial patterns in primes, utilizing advanced tools like a generalized von Neumann theorem and Gowers uniformity estimates.

Findings

Quantitative bounds match classical Siegel–Walfisz error terms.

First bounds for Tao–Ziegler polynomial patterns in primes.

Enhanced understanding of prime distribution along polynomial progressions.

Abstract

We prove quantitative estimates for averages of the von Mangoldt and M\"obius functions along polynomial progressions $n+P_1(m),\ldots, n+P_k(m)$ 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.