First Hardy Littlewood Conjecture
$begingroup$
The first Hardy Littlewood conjecture, also known as the k-Tuple conjecture is concisely presented here. However, I cannot find a paper explaining how Hardy and Littlewood came to such a conjecture. How is their statement justified? Where can the intuition behind the statement be understood? What paper presents a clear introduction to the conjecture and how it arose?
number-theory reference-request prime-numbers
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
$endgroup$
add a comment |
$begingroup$
The first Hardy Littlewood conjecture, also known as the k-Tuple conjecture is concisely presented here. However, I cannot find a paper explaining how Hardy and Littlewood came to such a conjecture. How is their statement justified? Where can the intuition behind the statement be understood? What paper presents a clear introduction to the conjecture and how it arose?
number-theory reference-request prime-numbers
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
$endgroup$
$begingroup$
Perhaps useful: mathoverflow.net/questions/54223/whence-the-k-tuple-conjecture
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Also: mathoverflow.net/questions/52700/…
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
add a comment |
$begingroup$
The first Hardy Littlewood conjecture, also known as the k-Tuple conjecture is concisely presented here. However, I cannot find a paper explaining how Hardy and Littlewood came to such a conjecture. How is their statement justified? Where can the intuition behind the statement be understood? What paper presents a clear introduction to the conjecture and how it arose?
number-theory reference-request prime-numbers
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
$endgroup$
The first Hardy Littlewood conjecture, also known as the k-Tuple conjecture is concisely presented here. However, I cannot find a paper explaining how Hardy and Littlewood came to such a conjecture. How is their statement justified? Where can the intuition behind the statement be understood? What paper presents a clear introduction to the conjecture and how it arose?
number-theory reference-request prime-numbers
number-theory reference-request prime-numbers
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
edited Jul 9 '17 at 0:06
Romain S
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
asked Jun 7 '17 at 3:02
Romain SRomain S
1,275722
1,275722
$begingroup$
Perhaps useful: mathoverflow.net/questions/54223/whence-the-k-tuple-conjecture
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Also: mathoverflow.net/questions/52700/…
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
add a comment |
$begingroup$
Perhaps useful: mathoverflow.net/questions/54223/whence-the-k-tuple-conjecture
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Also: mathoverflow.net/questions/52700/…
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Perhaps useful: mathoverflow.net/questions/54223/whence-the-k-tuple-conjecture
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Perhaps useful: mathoverflow.net/questions/54223/whence-the-k-tuple-conjecture
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Also: mathoverflow.net/questions/52700/…
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Also: mathoverflow.net/questions/52700/…
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
As pointed out in the comment section, you can read the article "Linear Equations in Primes" by Green and Tao, available on the ArXiv here:
http://arxiv.org/abs/math/0606088
in which they mention the works of Dickson as instrumental in the conception of such conjectures. In particular, you might be interested in reference [12].
In general, the intuition behind such conjectures involving specific sets of prime numbers is to observe their asymptotic behaviour using similar arguments as for all the primes, then use other arguments specific to the set in question to refine that asymptotic. For prime $k$-tuples, we take into account the number of open residue classes relative to the primes in each constellation in order to conjure up a multiplicative constant, similar to the twin-prime constant, that represents these residue classes.
In particular, given a prime constellation with $k$ members denoted by $P_k$ and a positive integer $n$,
$$
pi_{P_k}(n) sim C_{P_k} int _{2}^{n}{dt over (log t)^{k}},
$$
where $pi_{P_k}(n)$ denotes the amount of primes $pleq n$ such that $(p,p+ldots)in P_k$ and $C_{P_k}$ is the constant computed using the various open residue classes relative to the constellation.
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2312988%2ffirst-hardy-littlewood-conjecture%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
As pointed out in the comment section, you can read the article "Linear Equations in Primes" by Green and Tao, available on the ArXiv here:
http://arxiv.org/abs/math/0606088
in which they mention the works of Dickson as instrumental in the conception of such conjectures. In particular, you might be interested in reference [12].
In general, the intuition behind such conjectures involving specific sets of prime numbers is to observe their asymptotic behaviour using similar arguments as for all the primes, then use other arguments specific to the set in question to refine that asymptotic. For prime $k$-tuples, we take into account the number of open residue classes relative to the primes in each constellation in order to conjure up a multiplicative constant, similar to the twin-prime constant, that represents these residue classes.
In particular, given a prime constellation with $k$ members denoted by $P_k$ and a positive integer $n$,
$$
pi_{P_k}(n) sim C_{P_k} int _{2}^{n}{dt over (log t)^{k}},
$$
where $pi_{P_k}(n)$ denotes the amount of primes $pleq n$ such that $(p,p+ldots)in P_k$ and $C_{P_k}$ is the constant computed using the various open residue classes relative to the constellation.
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
$endgroup$
add a comment |
$begingroup$
As pointed out in the comment section, you can read the article "Linear Equations in Primes" by Green and Tao, available on the ArXiv here:
http://arxiv.org/abs/math/0606088
in which they mention the works of Dickson as instrumental in the conception of such conjectures. In particular, you might be interested in reference [12].
In general, the intuition behind such conjectures involving specific sets of prime numbers is to observe their asymptotic behaviour using similar arguments as for all the primes, then use other arguments specific to the set in question to refine that asymptotic. For prime $k$-tuples, we take into account the number of open residue classes relative to the primes in each constellation in order to conjure up a multiplicative constant, similar to the twin-prime constant, that represents these residue classes.
In particular, given a prime constellation with $k$ members denoted by $P_k$ and a positive integer $n$,
$$
pi_{P_k}(n) sim C_{P_k} int _{2}^{n}{dt over (log t)^{k}},
$$
where $pi_{P_k}(n)$ denotes the amount of primes $pleq n$ such that $(p,p+ldots)in P_k$ and $C_{P_k}$ is the constant computed using the various open residue classes relative to the constellation.
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
$endgroup$
add a comment |
$begingroup$
As pointed out in the comment section, you can read the article "Linear Equations in Primes" by Green and Tao, available on the ArXiv here:
http://arxiv.org/abs/math/0606088
in which they mention the works of Dickson as instrumental in the conception of such conjectures. In particular, you might be interested in reference [12].
In general, the intuition behind such conjectures involving specific sets of prime numbers is to observe their asymptotic behaviour using similar arguments as for all the primes, then use other arguments specific to the set in question to refine that asymptotic. For prime $k$-tuples, we take into account the number of open residue classes relative to the primes in each constellation in order to conjure up a multiplicative constant, similar to the twin-prime constant, that represents these residue classes.
In particular, given a prime constellation with $k$ members denoted by $P_k$ and a positive integer $n$,
$$
pi_{P_k}(n) sim C_{P_k} int _{2}^{n}{dt over (log t)^{k}},
$$
where $pi_{P_k}(n)$ denotes the amount of primes $pleq n$ such that $(p,p+ldots)in P_k$ and $C_{P_k}$ is the constant computed using the various open residue classes relative to the constellation.
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
$endgroup$
As pointed out in the comment section, you can read the article "Linear Equations in Primes" by Green and Tao, available on the ArXiv here:
http://arxiv.org/abs/math/0606088
in which they mention the works of Dickson as instrumental in the conception of such conjectures. In particular, you might be interested in reference [12].
In general, the intuition behind such conjectures involving specific sets of prime numbers is to observe their asymptotic behaviour using similar arguments as for all the primes, then use other arguments specific to the set in question to refine that asymptotic. For prime $k$-tuples, we take into account the number of open residue classes relative to the primes in each constellation in order to conjure up a multiplicative constant, similar to the twin-prime constant, that represents these residue classes.
In particular, given a prime constellation with $k$ members denoted by $P_k$ and a positive integer $n$,
$$
pi_{P_k}(n) sim C_{P_k} int _{2}^{n}{dt over (log t)^{k}},
$$
where $pi_{P_k}(n)$ denotes the amount of primes $pleq n$ such that $(p,p+ldots)in P_k$ and $C_{P_k}$ is the constant computed using the various open residue classes relative to the constellation.
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
answered Dec 20 '18 at 9:51
KlangenKlangen
1,70511334
1,70511334
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2312988%2ffirst-hardy-littlewood-conjecture%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Perhaps useful: mathoverflow.net/questions/54223/whence-the-k-tuple-conjecture
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21
$begingroup$
Also: mathoverflow.net/questions/52700/…
$endgroup$
– Matthew Conroy
Jun 7 '17 at 6:21