- journal article metadata
European Mathematical Society Publishing House
2017-10-05 23:40:02
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
33
2017
3
Clusters of primes with square-free translates
Roger
Baker
Brigham Young University, Provo, USA
Paul
Pollack
University of Georgia, Athens, USA
Maynard–Tao method, primes with square-free translates, mixed exponential sums
Let $\mathcal R$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal R$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{{7}/{12}+\epsilon}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in ({8}/{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+\epsilon}$.
Number theory
809
829
10.4171/RMI/956
http://www.ems-ph.org/doi/10.4171/RMI/956