Abstract
In a recent and much celebrated breakthrough, Maynard and Tao have independently proved a certain approximation to the prime k-tuple conjecture. We have subsequently seen numerous interesting applications of the Maynard–Tao sieve method, and in this short survey we will discuss some of its consequences for patterns in the gaps between consecutive primes. These include a conjecture of Erdős and Turán (as pointed out by Granville), and a conjecture of Chowla (first proved by Shiu in 1997). More recently it has been realized that the Maynard–Tao sieve method does not only produce small gaps between primes, but, as we will also discuss, it may be combined with a construction of Erdős–Rankin for producing large gaps between consecutive primes, resulting in small, medium and large gaps between consecutive primes.
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Notes
- 1.
Their condition being that the primes have level of distribution greater than 1∕2 (see (10) below).
- 2.
Maynard [21] writes: “Terence Tao (private communication) has independently proven Theorem 1.1 […] at much the same time.”
- 3.
Private communication.
- 4.
We used the ad hoc construction of the original version of this note: if \(\ell_{1} < \cdots <\ell _{k}\) are primes such that \(k <\ell _{1}\) and \(\ell_{k} <\ell_{ 1}^{2}\), and if \(Q = (\ell_{1}\cdots \ell_{k})^{-1}\prod _{p\leq \ell_{k}}p\), then \(\{Qx +\ell _{i}\}_{i=1}^{k}\) is admissible, and whenever it produces primes they must be consecutive. To guarantee the existence of the k primes \(\ell_{i}\) with \(\ell_{k} <\ell_{ 1}^{2}\), we applied the Maynard–Tao theorem to an admissible K-tuple with K sufficiently large in terms of k.
- 5.
See terrytao.wordpress.com/2013/11/22/ polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-254079.
- 6.
In fact the number of solutions is k, because \(p \nmid U\) implies \(p \nmid g_{1}\cdots g_{k}\prod _{1\leq i<j\leq k}(g_{i}h_{j} - g_{j}h_{i})\), so we have \(\prod _{i=1}^{k}(g_{i}n + h_{i}) \equiv 0\bmod p\) if and only if \(n \equiv -g_{i}^{-1}h_{i}\bmod p\) for some i, and these k congruence classes are all distinct mod p.
- 7.
In fact, a level of distribution of \(\theta = 0.971\) suffices.
- 8.
Let us write \(\mathcal{H} =\{ h_{1},\ldots,h_{k}\}\) now, instead of \(\mathcal{H}(x) =\{ x + h_{1},\ldots,x + h_{k}\}\) as in Sect. 2.
- 9.
Actually, we need to restrict the sum to moduli r W that avoid multiples of any “exceptional moduli” arising from putative Siegel zeros—a technical complication, which we ignore here for the sake of exposition.
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Acknowledgements
For their input or commentary, we thank William Banks, Andrew Granville, James Maynard, Caroline Turnage-Butterbaugh, and the referee.
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Freiberg, T. (2015). A Note on the Theorem of Maynard and Tao. In: Alaca, A., Alaca, Ş., Williams, K. (eds) Advances in the Theory of Numbers. Fields Institute Communications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3201-6_4
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