Asymptotics for some polynomial patterns in the primes
Pierre-Yves Bienvenu

TL;DR
This paper develops asymptotic formulas for sums involving polynomial patterns in primes, unifying previous results and enabling analysis of prime configurations like arithmetic progressions with special differences.
Contribution
It introduces a new pseudorandom majorant that unifies the treatment of the von Mangoldt function and quadratic form representation functions, extending asymptotic analysis to polynomial prime patterns.
Findings
Asymptotics for prime k-term arithmetic progressions with sum-of-two-squares differences
Unified approach to von Mangoldt and quadratic form functions
Extension of Green-Tao and Matthiesen results to polynomial patterns in primes
Abstract
We prove asymptotic formulae for sums of the form where is a convex body, each is either the von Mangoldt function or the representation function of a quadratic form, and is a system of linear forms of finite complexity. When all the functions are equal to the von Mangoldt function, we recover a result of Green and Tao, while when they are all representation functions of quadratic forms, we recover a result of Matthiesen. Our formulae imply asymptotics for some polynomial patterns in the primes. Specifically, they describe the asymptotic behaviour of the number of -term arithmetic progressions of primes whose common difference is a sum of two squares. The article combines ingredients from the work of Green and Tao on linear equations in primes and that of Matthiesen on…
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