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 any [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. The precise definition of the [math]\displaystyle{ MPZ }[/math], [math]\displaystyle{ MPZ' }[/math] and [math]\displaystyle{ MPZ'' }[/math] estimates can be found at the page on distribution of primes in smooth moduli.

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_0-2}^2}{k_0(k_0-1)} }[/math]

was obtained, where [math]\displaystyle{ j_{k_0-2}=j_{k_0-2,1} }[/math] is the first positive zero of the Bessel function [math]\displaystyle{ J_{k_0-2} }[/math]. See for instance this post for details. Note that even if the full Elliott-Halberstam conjecture was obtained, one could not reduce [math]\displaystyle{ k_0 }[/math] to below 6 by this criterion (and similarly for the variants of this implication given below), so [math]\displaystyle{ DHL[6,2] }[/math] is the limit of this method.

Converting MPZ to DHL

The observation that [math]\displaystyle{ DHL[k_0,2] }[/math] could be deduced from [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] if [math]\displaystyle{ k_0 }[/math] was sufficiently large depending on [math]\displaystyle{ \varpi,\delta }[/math] was first made in the literature by Motohashi and Pintz. In the paper of Zhang, an explicit implication was established: [math]\displaystyle{ MPZ[\varpi,\varpi] }[/math] implies [math]\displaystyle{ DHL[k_0,2] }[/math] whenever there exists an integer [math]\displaystyle{ l_0\gt 0 }[/math] such that

[math]\displaystyle{ (1+4\varpi) (1-\kappa_2) \gt (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa_1) }[/math]

where

[math]\displaystyle{ \kappa_1 := \delta_1( 1 + \delta_2^2 + k_0 \log(1+\frac{1}{4\varpi}) \binom{k_0+2l_0}{k_0} }[/math]

[math]\displaystyle{ \kappa_2 := \delta_1 (1+4\varpi) ( 1 + \delta_2^2 + k_0 \log(1+\frac{1}{4\varpi}) \binom{k_0+2l_0+1}{k_0-1} }[/math]

[math]\displaystyle{ \delta_1 := (1+4\varpi)^{-k_0} }[/math]

[math]\displaystyle{ \delta_2 := \sum_{0 \leq j \lt 1+\frac{1}{4\varpi}} \frac{ \log(1+\frac{1}{4\varpi}) k_0)^j}{j!}. }[/math]

The value of [math]\displaystyle{ \delta_2 }[/math] was lowered to [math]\displaystyle{ \prod_{0 \leq j \lt 1+\frac{1}{4\varpi}} (1 + k_0 \log(1+\frac{1}{j}) }[/math] in these notes. Subsequently, the values of [math]\displaystyle{ \kappa_1,\kappa_2 }[/math] were improved to

[math]\displaystyle{ \kappa_1 := (\delta_1 + \sum_{j=1}^{1/4\varpi} \delta_1^j \delta_{2,j}^2 + \delta_1 k_0 \log(1+\frac{1}{4\varpi})) \binom{k_0+2l_0}{k_0} }[/math]

[math]\displaystyle{ \kappa_2 := (\delta_1 (1+4\varpi) + \sum_{j=1}^{1/4\varpi} \delta_1^j (1+4\varpi)^j \delta_{2,j}^2 + \delta_1 (1+4\varpi) k_0 \log(1+\frac{1}{4\varpi}) \binom{k_0+2l_0+1}{k_0-1} }[/math]

where

[math]\displaystyle{ \delta_{2,j} := \prod_{i=1}^j (1 + k_0 \log(1+\frac{1}{i}) ) }[/math];

again, see these notes. As before, [math]\displaystyle{ l_0 }[/math] can be taken to be non-integer.

The constraint was then improved further in this post to deduce [math]\displaystyle{ DHL[k_0,2] }[/math] from [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] whenever

[math]\displaystyle{ (1+4\varpi) \gt (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa) }[/math]

where

[math]\displaystyle{ \kappa = \sum_{1 \leq n \leq \frac{1+4\varpi}{2\delta}} (1 - \frac{2n\delta}{1+4\varpi})^{k_0/2+l_0} \prod_{j=1}^n (1+3k_0 \log(1+\frac{1}{j})) }[/math].

Using the optimal Bessel weight, this condition was improved to

[math]\displaystyle{ (1+4\varpi) \gt \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} (1 + \kappa) }[/math];

again, see this post.

A variant of this criterion was developed using the elementary Selberg sieve in this post, but never used. A subsequent refined criterion was established in this post, namely that

[math]\displaystyle{ (1+4\varpi) (1-\kappa') \gt \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} (1 + \kappa) }[/math]

where

[math]\displaystyle{ \kappa := \sum_{1 \leq n \lt \frac{1+4\varpi}{2\delta}} \frac{3^n+1}{2} \frac{k_0^n}{n!} (\int_{4\delta/(1+\varpi)}^1 (1-t)^{k_0/2} \frac{dt}{t})^n }[/math]

[math]\displaystyle{ \kappa' := \sum_{2 \leq n \lt \frac{1+4\varpi}{2\delta}} \frac{3^n-1}{2} \frac{(k_0-1)^n}{n!} (\int_{4\delta/(1+\varpi)}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t})^n. }[/math]

A slight refinement in this comment allows the condition [math]\displaystyle{ n \geq 2 }[/math] in the definition of [math]\displaystyle{ \kappa' }[/math] to be raised to [math]\displaystyle{ n \geq 3 }[/math].

An argument of Pintz yields the following improved values of [math]\displaystyle{ \kappa,\kappa' }[/math] in the above criterion:

[math]\displaystyle{ \kappa := 0 }[/math]

[math]\displaystyle{ \kappa' := 2 \kappa_1 + 2 \kappa_2 }[/math]

[math]\displaystyle{ \kappa_1 := \int_{4\delta/(1+4\varpi)}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t} }[/math]

[math]\displaystyle{ \kappa_2 := (k_0-1) \int_{4\delta/(1+4\varpi)}^1 (1-t)^{k_0-1} \frac{dt}{t} }[/math].

Converting MPZ' to DHL

An efficient argument of Pintz, based on the elementary Selberg sieve, allows one to deduce [math]\displaystyle{ DHL[k_0,2] }[/math] from [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math] with almost no loss with respect to the [math]\displaystyle{ \delta }[/math] parameter. As currently optimised, the criterion takes the form

[math]\displaystyle{ (1+4\varpi) (1-2\kappa_1 - 2\kappa_2 - 2\kappa_3) \gt \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} }[/math]

where

[math]\displaystyle{ \kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t} }[/math]

[math]\displaystyle{ \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t} }[/math]

[math]\displaystyle{ \kappa_3 := \tilde \theta \frac{J_{k_0-2}(\sqrt{\tilde \theta} j_{k_0-2})^2 - J_{k_0-3}(\sqrt{\tilde \theta} j_{k_0-2}) J_{k_0-1}(\sqrt{\tilde \theta} j_{k_0-2})}{ J_{k_0-3}(j_{k_0-2})^2 } \exp( A + (k_0-1) \int_{\tilde \delta}^\theta e^{-(A+2\alpha)t} \frac{dt}{t} ) }[/math]

[math]\displaystyle{ \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)} }[/math]

[math]\displaystyle{ \theta := \frac{\delta'}{1/4 + \varpi} }[/math]

[math]\displaystyle{ \tilde \theta := \frac{(\delta' - \delta)/2 + \varpi}{1/4 + \varpi} }[/math]

[math]\displaystyle{ \tilde \delta := \frac{\delta}{1/4 + \varpi} }[/math]

and [math]\displaystyle{ A\gt 0 }[/math] and [math]\displaystyle{ \delta \leq \delta' \leq \frac{1}{4} + \varpi }[/math] are parameters one is free to optimise over.

Here is some simple Maple code to verify the above criterion for given choices of [math]\displaystyle{ k_0,\varpi,\delta,\delta',A }[/math]:

k0 := [INSERT VALUE HERE];
varpi := [INSERT VALUE HERE];
delta := [INSERT VALUE HERE];
deltap := [INSERT VALUE HERE]; 
A := [INSERT VALUE HERE];
theta := deltap / (1/4 + varpi);
thetat := ((deltap - delta)/2 + varpi) / (1/4 + varpi);
deltat := delta / (1/4 + varpi);
j := BesselJZeros(k0-2,1);
eps := 1 - j^2 / (k0 * (k0-1) * (1+4*varpi));
kappa1 := int( (1-t)^((k0-1)/2)/t, t = theta..1, numeric);
kappa2 := (k0-1) * int( (1-t)^(k0-1)/t, t=theta..1, numeric);
alpha := j^2 / (4 * (k0-1));
e := exp( A + (k0-1) * int( exp(-(A+2*alpha)*t)/t, t=deltat..theta, numeric ) );
gd := (j^2/2) * BesselJ(k0-3,j)^2;
tn := sqrt(thetat)*j;
gn := (tn^2/2) * (BesselJ(k0-2,tn)^2 - BesselJ(k0-3,tn)*BesselJ(k0-1,tn));
kappa3 := (gn/gd) * e;
eps2 := 2*(kappa1+kappa2+kappa3);
# we win if eps2 < eps

Converting [math]\displaystyle{ MPZ'' }[/math] to DHL

There is a variant of [math]\displaystyle{ MPZ' }[/math] which we call [math]\displaystyle{ MPZ'' }[/math] in which dense divisibility is replaced by the stronger condition of double dense divisibility; see [Distribution of primes in smooth moduli] for details. [math]\displaystyle{ MPZ''[\varpi,\delta] }[/math] is weaker than [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math] but stronger than [math]\displaystyle{ MPZ[\varpi,\delta] }[/math]. It turns out (details here) that one can also deduce [math]\displaystyle{ DHL[k_0,2] }[/math] from [math]\displaystyle{ MPZ''[\varpi,\delta] }[/math] with an almost identical numerology to the previous section, except that [math]\displaystyle{ \tilde \theta }[/math] is increased to

[math]\displaystyle{ \tilde \theta := \min( \frac{\delta' - \delta/2 + \varpi}{1/4 + \varpi}, 1). }[/math]

So the Maple code is now changed slightly to the following:

k0 := [INSERT VALUE HERE];
varpi := [INSERT VALUE HERE];
delta := [INSERT VALUE HERE];
deltap := [INSERT VALUE HERE]; 
A := [INSERT VALUE HERE];
theta := deltap / (1/4 + varpi);
thetat := min( ((deltap - delta/2) + varpi) / (1/4 + varpi), 1);
deltat := delta / (1/4 + varpi);
j := BesselJZeros(k0-2,1);
eps := 1 - j^2 / (k0 * (k0-1) * (1+4*varpi));
kappa1 := int( (1-t)^((k0-1)/2)/t, t = theta..1, numeric);
kappa2 := (k0-1) * int( (1-t)^(k0-1)/t, t=theta..1, numeric);
alpha := j^2 / (4 * (k0-1));
e := exp( A + (k0-1) * int( exp(-(A+2*alpha)*t)/t, t=deltat..theta, numeric ) );
gd := (j^2/2) * BesselJ(k0-3,j)^2;
tn := sqrt(thetat)*j;
gn := (tn^2/2) * (BesselJ(k0-2,tn)^2 - BesselJ(k0-3,tn)*BesselJ(k0-1,tn));
kappa3 := (gn/gd) * e;
eps2 := 2*(kappa1+kappa2+kappa3);
# we win if eps2 < eps

More generally, to deduce [math]\displaystyle{ DHL[k_0,2] }[/math] from [math]\displaystyle{ MPZ^{(k)}[\varpi,\delta] }[/math], we need to use

[math]\displaystyle{ \tilde \theta := \min( \frac{k\delta'/2 - \delta/2 + \varpi}{1/4 + \varpi}, 1). }[/math]