Dickson-Hardy-Littlewood theorems

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:<math> \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 }  
:<math> \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 }  
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\exp( A + (k0-1) \int_{\tilde \delta}^\theta e^{-(A+2\alpha)t} \frac{dt}{t} )</math>
+
\exp( A + (k_0-1) \int_{\tilde \delta}^\theta e^{-(A+2\alpha)t} \frac{dt}{t} )</math>
:<math> \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}</math>
:<math> \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}</math>

Revision as of 07:04, 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. The precise definition of the MPZ and MPZ' 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 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_0-2}^2}{k_0(k_0-1)}

was obtained, where j_{k_0-2}=j_{k_0-2,1} is the first positive zero of the Bessel function J_{k_0-2}. See for instance this post for details.

Converting MPZ to DHL

The observation that DHL[k0,2] could be deduced from MPZ[\varpi,\delta] if k0 was sufficiently large depending on \varpi,\delta was first made in the literature by Motohashi and Pintz. In the paper of Zhang, an explicit implication was established: MPZ[\varpi,\varpi] implies DHL[k0,2] whenever there exists an integer l0 > 0 such that

 (1+4\varpi) (1-\kappa_2) > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa_1)

where

 \kappa_1 := \delta_1( 1 + \delta_2^2 + k_0 \log(1+\frac{1}{4\varpi}) \binom{k_0+2l_0}{k_0}
 \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}
 \delta_1 := (1+4\varpi)^{-k_0}
 \delta_2 := \sum_{0 \leq j < 1+\frac{1}{4\varpi}} \frac{ \log(1+\frac{1}{4\varpi}) k_0)^j}{j!}.

The value of δ2 was lowered to \prod_{0 \leq j < 1+\frac{1}{4\varpi}} (1 + k_0 \log(1+\frac{1}{j}) in these notes. Subsequently, the values of κ12 were improved to

 \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}
 \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}

where

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

again, see these notes. As before, l0 can be taken to be non-integer.

The constraint was then improved further in this post to deduce DHL[k0,2] from MPZ[\varpi,\delta] whenever

 (1+4\varpi) > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa)

where

 \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})).

Using the optimal Bessel weight, this condition was improved to

 (1+4\varpi) > \frac{j_{k_0-2}}{k_0(k_0-1)} (1 + \kappa);

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

 (1+4\varpi) (1-\kappa') > \frac{j_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)

where

 \kappa := \sum_{1 \leq n < \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
 \kappa' := \sum_{2 \leq n < \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.

A slight refinement in this comment allows the condition n \geq 2 in the definition of κ' to be raised to n \geq 3.

An argument of Pintz yields the following improved values of κ,κ' in the above criterion:

κ: = 0
κ': = 2κ1 + 2κ2
 \kappa_1 := \int_{4\delta/(1+4\varpi)}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t}
 \kappa_2 := (k_0-1) \int_{4\delta/(1+4\varpi)}^1 (1-t)^{k_0-1} \frac{dt}{t}.

Converting MPZ' to DHL

An efficient argument of Pintz, based on the elementary Selberg sieve, allows one to deduce DHL[k0,2] from MPZ'[\varpi,\delta] with almost no loss with respect to the δ parameter. As currently optimised, the criterion takes the form

 (1+4\varpi) (1-2\kappa_1 - 2\kappa_2 - 2\kappa_3) > \frac{j_{k_0-2}}{k_0(k_0-1)}

where

 \kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t}
 \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t}
 \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} )
 \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}
 \theta := \frac{\delta'}{1/4 + \varpi}
 \tilde \theta := \frac{(\delta' - \delta)/2 + \varpi}{1/4 + \varpi}
 \tilde \delta := \frac{\delta}{1/4 + \varpi}

and A > 0 and \delta \leq \delta' \leq \frac{1}{4} + \varpi are parameters one is free to optimise over.

Here is some simple Maple code to verify the above criterion for given choices of k_0,\varpi,\delta,\delta',A:

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
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