Dickson-Hardy-Littlewood theorems
For any integer [math]\displaystyle{ k_0 \geq 2 }[/math], let [math]\displaystyle{ DHL[k_0,2] }[/math] denote the assertion that given any admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ {\mathcal H} }[/math], that infinitely many translates of [math]\displaystyle{ {\mathcal H} }[/math] contain at least two primes. Thus for instance [math]\displaystyle{ DHL[2,2] }[/math] 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 [math]\displaystyle{ DHL[k_0,2] }[/math] from three classes of estimates:
- Elliott-Halberstam estimates [math]\displaystyle{ EH[\theta] }[/math] for some [math]\displaystyle{ 1/2 \lt \theta \lt 1 }[/math].
- Motohashi-Pintz-Zhang estimates [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] for some [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math] and [math]\displaystyle{ 0 \lt \delta \lt 1/4+\varpi }[/math].
- Motohashi-Pintz-Zhang estimates [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math] for densely divisible moduli for some [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math] and [math]\displaystyle{ 0 \lt \delta \lt 1/4+\varpi }[/math].
The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form [math]\displaystyle{ EH[\theta] }[/math] for nay [math]\displaystyle{ \theta \gt 1/2 }[/math] is known unconditionally at present. Zhang was the first to establish a result of the form [math]\displaystyle{ MPZ[\varpi,\theta] }[/math], which is weaker than [math]\displaystyle{ EH[1/2+2\varpi] }[/math], for some [math]\displaystyle{ \varpi,\theta\gt 0 }[/math]. More recently, we have switched to using [math]\displaystyle{ MPZ'[\varpi,\theta] }[/math], an estimate of intermediate strength between [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] and [math]\displaystyle{ EH[1/2+2\varpi] }[/math], as the conversion of this estimate to a [math]\displaystyle{ DHL[k_0,2] }[/math] result is more efficient in the [math]\displaystyle{ \delta }[/math] parameter.
Converting EH to DHL
In the breakthrough paper of Goldston, Pintz, and Yildirim, it was shown that [math]\displaystyle{ EH[\theta] }[/math] implied [math]\displaystyle{ DHL[k_0,2] }[/math] whenever
- [math]\displaystyle{ 2\theta \gt (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) }[/math]
for some positive integer [math]\displaystyle{ l_0 }[/math]. Actually (as noted here), there is nothing preventing the argument for working for non-integer [math]\displaystyle{ l_0 \gt 0 }[/math] as well, so we can optimise this condition as
- [math]\displaystyle{ 2\theta \gt (1 + \frac{1}{\sqrt{k_0}})^2 }[/math].
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
- [math]\displaystyle{ 2\theta \gt \frac{j_{k_0-1}^2}{k_0(k_0-1)} }[/math]
was obtained, where [math]\displaystyle{ j_{k_0-1}=j_{k_0-1,1} }[/math] is the first positive zero of the Bessel function [math]\displaystyle{ J_{k_0-1} }[/math]. See for instance this post for details.
