Recent zbMATH articles in MSC 11N32 https://zbmath.org/atom/cc/11N32 2021-05-28T16:06:00+00:00 Werkzeug Polynomial patterns in the primes. https://zbmath.org/1459.11030 2021-05-28T16:06:00+00:00 "Tao, Terence" https://zbmath.org/authors/?q=ai:tao.terence-c "Ziegler, Tamar" https://zbmath.org/authors/?q=ai:ziegler.tamar After the celebrated result of the first author with \textit{B. Green} concerning the existence of arbitrarily long progressions of primes [Ann. Math. (2) 167, No. 2, 481--547 (2008; Zbl 1191.11025)], several variations involving polynomial patterns in the primes appeared. For example, in [\textit{T. Tao} and \textit{T. Ziegler}, Acta Math. 201, No. 2, 213--305 (2008; Zbl 1230.11018)] it is shown that if $$P_1,\ldots,P_k\in {\mathbb Z}[m]$$ are polynomials in one indeterminate with $$P_1(0)=\cdots = P_k(0) = 0$$, then there are infinitely many pairs $$(m,n)$$ such that $$n+P_i(m)$$ are primes for $$i=1,\ldots,k$$. In this paper, the authors formulate and prove various generalizations involving polynomials of several variables satisfying a technical degree condition. That is, let $$d$$ and $$r$$ be natural numbers and $$P_1,\ldots,P_k\in {\mathbb Z}[m_1,\ldots,m_r]$$ are polynomials of $$r$$ variables, degree at most $$d$$ such that the degree $$d$$ components of $$P_1,\ldots,P_k$$ are all distinct (that is $$P_i-P_j$$ has degree exactly $$d$$ for all $$1\le i\ne j\le k$$). Assume further that for each prime $$p$$ there are $$n$$ in $${\vec{m}}\in {\mathbb Z}[m_1,\ldots,m_r]$$ such that none of $$n+P_1(\vec{m}),\ldots,n+P_k(\vec{m})$$ are multiples of $$p$$. Then there are infinitely many natural numbers $$n,m_1,\ldots,m_r$$ such that $$n+P_1(m_1,\ldots,m_r),\ldots,n+P_k(m_1,\ldots,m_r)$$ are all primes. For small $$d$$ the authors can eliminate the condition that the $$d$$ components of $$P_1,\ldots,P_k$$ are all distinct (see Theorem 5 of the paper). The authors also present quantitative versions of this result and narrow polynomial patterns in the subsets of the primes. For example, let $${\mathcal A}$$ be a subset of primes of positive upper density and $$P_1,\ldots,P_k\in {\mathbb Z}[m]$$ be polynomials of degree at most $$d$$ with $$P_1(0)=\cdots=P_k(0)$$. Then one can find infinitely many natural numbers $$n$$ and $$m$$ such that $$n+P_1(m),\ldots,n+P_k(m)$$ are all primes in $${\mathcal A}$$ and further $$m\ll (\log n)^L$$, where $$L$$ depends only on $$d$$ and $$k$$. The proof is very technical and uses the Cauchy-Schwartz inequality and the circle method to control the average Gowers norms of what the authors call a $$W$$-tricked von Mangoldt function, which is the von Mangoldt function of integers coprime to $$W$$ normalised by the factor $$\phi(W)/W$$, and $$W=\prod_{p\le w} p$$ for an appropriate value of the sifting parameter $$w$$. Reviewer: Florian Luca (Johannesburg)