Recent zbMATH articles in MSC 11N32https://zbmath.org/atom/cc/11N322021-05-28T16:06:00+00:00WerkzeugPolynomial patterns in the primes.https://zbmath.org/1459.110302021-05-28T16:06:00+00:00"Tao, Terence"https://zbmath.org/authors/?q=ai:tao.terence-c"Ziegler, Tamar"https://zbmath.org/authors/?q=ai:ziegler.tamarAfter 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)