Polymath Wiki - User contributions [en]

Dickson-Hardy-Littlewood theorems

2013-07-01T20:25:03Z

Pace: Added a 2 to an exponent on a j.

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 translates of <math>{\mathcal H}</math> contain at least two primes. Thus for instance <math>DHL[2,2]</math> would imply the twin prime conjecture. The acronym DHL stands for "Dickson-Hardy-Littlewood", and originates from [http://arxiv.org/abs/1305.6289 this paper of Pintz].

It is known how to deduce results <math>DHL[k_0,2]</math> from three classes of estimates:

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

The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form <math>EH[\theta]</math> for nay <math>\theta > 1/2</math> is known unconditionally at present. Zhang was the first to establish a result of the form <math>MPZ[\varpi,\theta]</math>, which is weaker than <math>EH[1/2+2\varpi+]</math>, for some <math>\varpi,\theta>0</math>. More recently, we have switched to using <math>MPZ'[\varpi,\theta]</math>, an estimate of intermediate strength between <math>MPZ[\varpi,\delta]</math> and <math>EH[1/2+2\varpi+]</math>, as the conversion of this estimate to a <math>DHL[k_0,2]</math> result is more efficient in the <math>\delta</math> 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 [http://www.ams.org/mathscinet-getitem?mr=2552109 breakthrough paper of Goldston, Pintz, and Yildirim], it was shown that <math>EH[\theta]</math> implied <math>DHL[k_0,2]</math> whenever

:<math>2\theta > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0})</math>

for some positive integer <math>l_0</math>. Actually (as noted [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here]), there is nothing preventing the argument for working for non-integer <math>l_0 > 0</math> as well, so we can optimise this condition as

:<math>2\theta > (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 [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf this paper of Farkas, Pintz, and Revesz], the optimal weight was found (coming from a Bessel function), and the optimised condition

:<math>2\theta > \frac{j_{k_0-2}^2}{k_0(k_0-1)}</math>

was obtained, where <math>j_{k_0-2}=j_{k_0-2,1}</math> is the first positive zero of the Bessel function <math>J_{k_0-2}</math>. See for instance [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ this post] for details.

== Converting MPZ to DHL ==

The observation that <math>DHL[k_0,2]</math> could be deduced from <math>MPZ[\varpi,\delta]</math> if <math>k_0</math> was sufficiently large depending on <math>\varpi,\delta</math> was first made in the literature [http://www.ams.org/mathscinet-getitem?mr=2414788 by Motohashi and Pintz]. In the [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf paper of Zhang], an explicit implication was established: <math>MPZ[\varpi,\varpi]</math> implies <math>DHL[k_0,2]</math> whenever there exists an integer <math>l_0>0</math> such that

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

where

:<math> \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> \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> \delta_1 := (1+4\varpi)^{-k_0}</math>

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

The value of <math>\delta_2</math> was lowered to <math>\prod_{0 \leq j < 1+\frac{1}{4\varpi}} (1 + k_0 \log(1+\frac{1}{j})</math> in [http://terrytao.files.wordpress.com/2013/05/bounds.pdf these notes]. Subsequently, the values of <math>\kappa_1,\kappa_2</math> were improved to

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

again, see [http://terrytao.files.wordpress.com/2013/05/bounds.pdf these notes]. As before, <math>l_0</math> can be taken to be non-integer.

The constraint was then [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ improved further in this post] to deduce <math>DHL[k_0,2]</math> from <math>MPZ[\varpi,\delta]</math> whenever

:<math> (1+4\varpi) > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa)</math>

where

:<math> \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> (1+4\varpi) > \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)</math>;

again, see [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ this post].

A variant of this criterion was developed using the elementary Selberg sieve in [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ this post], but never used. A subsequent refined criterion was established in [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ this post], namely that

:<math> (1+4\varpi) (1-\kappa') > \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)</math>

where

:<math> \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</math>

:<math> \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.</math>

A slight refinement in [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234845 this comment] allows the condition <math>n \geq 2</math> in the definition of <math>\kappa'</math> to be raised to <math>n \geq 3</math>.

An [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz argument of Pintz] yields the following improved values of <math>\kappa,\kappa'</math> in the above criterion:

:<math> \kappa := 0 </math>

:<math> \kappa' := 2 \kappa_1 + 2 \kappa_2</math>

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

:<math> \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 [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz argument of Pintz], based on the elementary Selberg sieve, allows one to deduce <math>DHL[k_0,2]</math> from <math>MPZ'[\varpi,\delta]</math> with almost no loss with respect to the <math>\delta</math> parameter. As currently optimised, the criterion takes the form

:<math> (1+4\varpi) (1-2\kappa_1 - 2\kappa_2 - 2\kappa_3) > \frac{j^{2}_{k_0-2}}{k_0(k_0-1)}</math>

where

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

:<math> \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t}</math>

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

and <math>A>0</math> and <math>\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>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

Dickson-Hardy-Littlewood theorems

2013-07-01T20:24:14Z

Pace: /* Converting MPZ to DHL */ Added the exponent 2 to some of the j's

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 translates of <math>{\mathcal H}</math> contain at least two primes. Thus for instance <math>DHL[2,2]</math> would imply the twin prime conjecture. The acronym DHL stands for "Dickson-Hardy-Littlewood", and originates from [http://arxiv.org/abs/1305.6289 this paper of Pintz].

It is known how to deduce results <math>DHL[k_0,2]</math> from three classes of estimates:

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

The Elliott-Halberstam estimates are the simplest to use, but unfortunately no estimate of the form <math>EH[\theta]</math> for nay <math>\theta > 1/2</math> is known unconditionally at present. Zhang was the first to establish a result of the form <math>MPZ[\varpi,\theta]</math>, which is weaker than <math>EH[1/2+2\varpi+]</math>, for some <math>\varpi,\theta>0</math>. More recently, we have switched to using <math>MPZ'[\varpi,\theta]</math>, an estimate of intermediate strength between <math>MPZ[\varpi,\delta]</math> and <math>EH[1/2+2\varpi+]</math>, as the conversion of this estimate to a <math>DHL[k_0,2]</math> result is more efficient in the <math>\delta</math> 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 [http://www.ams.org/mathscinet-getitem?mr=2552109 breakthrough paper of Goldston, Pintz, and Yildirim], it was shown that <math>EH[\theta]</math> implied <math>DHL[k_0,2]</math> whenever

:<math>2\theta > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0})</math>

for some positive integer <math>l_0</math>. Actually (as noted [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ here]), there is nothing preventing the argument for working for non-integer <math>l_0 > 0</math> as well, so we can optimise this condition as

:<math>2\theta > (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 [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf this paper of Farkas, Pintz, and Revesz], the optimal weight was found (coming from a Bessel function), and the optimised condition

:<math>2\theta > \frac{j_{k_0-2}^2}{k_0(k_0-1)}</math>

was obtained, where <math>j_{k_0-2}=j_{k_0-2,1}</math> is the first positive zero of the Bessel function <math>J_{k_0-2}</math>. See for instance [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ this post] for details.

== Converting MPZ to DHL ==

The observation that <math>DHL[k_0,2]</math> could be deduced from <math>MPZ[\varpi,\delta]</math> if <math>k_0</math> was sufficiently large depending on <math>\varpi,\delta</math> was first made in the literature [http://www.ams.org/mathscinet-getitem?mr=2414788 by Motohashi and Pintz]. In the [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf paper of Zhang], an explicit implication was established: <math>MPZ[\varpi,\varpi]</math> implies <math>DHL[k_0,2]</math> whenever there exists an integer <math>l_0>0</math> such that

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

where

:<math> \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> \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> \delta_1 := (1+4\varpi)^{-k_0}</math>

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

The value of <math>\delta_2</math> was lowered to <math>\prod_{0 \leq j < 1+\frac{1}{4\varpi}} (1 + k_0 \log(1+\frac{1}{j})</math> in [http://terrytao.files.wordpress.com/2013/05/bounds.pdf these notes]. Subsequently, the values of <math>\kappa_1,\kappa_2</math> were improved to

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

again, see [http://terrytao.files.wordpress.com/2013/05/bounds.pdf these notes]. As before, <math>l_0</math> can be taken to be non-integer.

The constraint was then [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ improved further in this post] to deduce <math>DHL[k_0,2]</math> from <math>MPZ[\varpi,\delta]</math> whenever

:<math> (1+4\varpi) > (1 + \frac{1}{2l_0+1}) (1 + \frac{2l_0+1}{k_0}) (1 + \kappa)</math>

where

:<math> \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> (1+4\varpi) > \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)</math>;

again, see [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ this post].

A variant of this criterion was developed using the elementary Selberg sieve in [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ this post], but never used. A subsequent refined criterion was established in [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ this post], namely that

:<math> (1+4\varpi) (1-\kappa') > \frac{j^{2}_{k_0-2}}{k_0(k_0-1)} (1 + \kappa)</math>

where

:<math> \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</math>

:<math> \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.</math>

A slight refinement in [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234845 this comment] allows the condition <math>n \geq 2</math> in the definition of <math>\kappa'</math> to be raised to <math>n \geq 3</math>.

An [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz argument of Pintz] yields the following improved values of <math>\kappa,\kappa'</math> in the above criterion:

:<math> \kappa := 0 </math>

:<math> \kappa' := 2 \kappa_1 + 2 \kappa_2</math>

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

:<math> \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 [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz argument of Pintz], based on the elementary Selberg sieve, allows one to deduce <math>DHL[k_0,2]</math> from <math>MPZ'[\varpi,\delta]</math> with almost no loss with respect to the <math>\delta</math> parameter. As currently optimised, the criterion takes the form

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

where

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

:<math> \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t}</math>

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

and <math>A>0</math> and <math>\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>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