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Journal of the European Mathematical Society

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Volume 23, Issue 2, 2021, pp. 667–700
DOI: 10.4171/JEMS/1020

Published online: 2020-11-15

Long gaps in sieved sets

Kevin Ford[1], Sergey Konyagin[2], James Maynard[3], Carl B. Pomerance[4] and Terence Tao[5]

(1) University of Illinois at Urbana-Champaign, USA
(2) Steklov Mathematical Institute, Moscow, Russia
(3) University of Oxford, UK
(4) Dartmouth College, Hanover, USA
(5) University of California, Los Angeles, USA

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about 1 on average. We show that for sufficiently large $x$, the sifted set $\{n \in \mathbb{Z}: n (\mathrm {mod}\: p) \not \in I_p \: \mathrm {for \: all} p \leq x\}$ contains gaps of size $x (\log x)^{\delta}$ depends only on the densitiy of primes for which $I_p \neq \emptyset$. This improves on the "trivial'' bound of $\gg x$. As a consequence, for any non-constant polynomial $f: \mathbb{Z} \to \mathbb{Z}$ with positive leading coefficient, the set $\{n \leq X: f(n) \; \mathrm {composite}\}$ contains an interval of consecutive integers of length $\ge (\log X) (\log\log X)^{\delta}$ for sufficiently large $X$, where $\delta > 0$ depends only on the degree of $f$.

Keywords: Gaps, prime values of polynomials, sieves

Ford Kevin, Konyagin Sergey, Maynard James, Pomerance Carl, Tao Terence: Long gaps in sieved sets. J. Eur. Math. Soc. 23 (2021), 667-700. doi: 10.4171/JEMS/1020