Dickson-Hardy-Littlewood theorems

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(New page: For any integer <math>k_0 \geq 2</math>, let <math>DHL[k_0,2]</math> denote the assertion that given any admissible <math>k_0</math>-tuple <math>{\mathcal H}</math>, that infinitely many t...)
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Revision as of 05:51, 26 June 2013

For any integer k_0 \geq 2, let DHL[k0,2] denote the assertion that given any admissible k0-tuple {\mathcal H}, that infinitely many translates of {\mathcal H} contain at least two primes. Thus for instance DHL[2,2] would imply the twin prime conjecture. The acronym DHL stands for "Dickson-Hardy-Littlewood", and originates from this paper of Pintz.

It is known how to deduce results DHL[k0,2] from three classes of estimates:

  • Elliott-Halberstam estimates EH[θ] for some 1 / 2 < θ < 1.
  • Motohashi-Pintz-Zhang estimates MPZ[\varpi,\delta] for some 0 < \varpi < 1/4 and 0 < \delta < 1/4+\varpi.
  • Motohashi-Pintz-Zhang estimates MPZ'[\varpi,\delta] for densely divisible moduli for some 0 < \varpi < 1/4 and 0 < \delta < 1/4+\varpi.

The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form EH[θ] for nay θ > 1 / 2 is known unconditionally at present. Zhang was the first to establish a result of the form MPZ[\varpi,\theta], which is weaker than EH[1/2+2\varpi], for some \varpi,\theta>0. More recently, we have switched to using MPZ'[\varpi,\theta], an estimate of intermediate strength between MPZ[\varpi,\delta] and EH[1/2+2\varpi], as the conversion of this estimate to a DHL[k0,2] result is more efficient in the δ parameter.

Converting EH to DHL

In the breakthrough paper of Goldston, Pintz, and Yildirim, it was shown that EH[θ] implied DHL[k0,2] whenever

2\theta > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0})

for some positive integer l0. Actually (as noted here), there is nothing preventing the argument for working for non-integer l0 > 0 as well, so we can optimise this condition as

2\theta > (1 + \frac{1}{\sqrt{k_0}})^2.

Some further optimisation of this condition was performed in the paper of Goldston, Pintz, and Yildirim by working with general polynomial weights rather than monomial weights. In this paper of Farkas, Pintz, and Revesz, the optimal weight was found (coming from a Bessel function), and the optimised condition

2\theta > \frac{j_{k_0-1}^2}{k_0(k_0-1)}

was obtained, where j_{k_0-1}=j_{k_0-1,1} is the first positive zero of the Bessel function J_{k_0-1}. See for instance this post for details.

Converting MPZ to DHL

Converting MPZ' to DHL

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