Mixing
A probabilistic attempt to solve Riemann Hypothesis using Mertens function.
$begingroup$
I know that the following statement:
For every $epsilon>0$
$$M(N)=O(N^{0.5+epsilon})$$ is equivalent to Riemann Hypothesis (Where $M(N)$ is Mertens function).
As Mertens function behaves somehow randomly here is idea to treat this function as a translation in random walk. Each step is related to values of independent random variable which is equal to
$$-1,0,+1$$
with probability respectively equal to
$$frac{3}{pi^{2}},1-frac{6}{pi^{2}},frac{3}{pi^{2}}$$
I know that the behaviour of translation will be more generally but on the other hand i saw a statement that given values of probability"uniquely determine the asymptotic behavior of the Mertens function"
I quoted it from this:
https://arxiv.org/ftp/arxiv/papers/1712/1712.04674.pdf
Does have it sense to use here probabilistic tools such as central limit theorem or law of iterated logarithm?
Thanks in advance for answers.
probability-theory asymptotics random-walk riemann-hypothesis
asked Jan 6 at 14:31
mkultramkultra
466
$endgroup$
add a comment |
$begingroup$
I know that the following statement:
For every $epsilon>0$
$$M(N)=O(N^{0.5+epsilon})$$ is equivalent to Riemann Hypothesis (Where $M(N)$ is Mertens function).
As Mertens function behaves somehow randomly here is idea to treat this function as a translation in random walk. Each step is related to values of independent random variable which is equal to
$$-1,0,+1$$
with probability respectively equal to
$$frac{3}{pi^{2}},1-frac{6}{pi^{2}},frac{3}{pi^{2}}$$
I know that the behaviour of translation will be more generally but on the other hand i saw a statement that given values of probability"uniquely determine the asymptotic behavior of the Mertens function"
I quoted it from this:
https://arxiv.org/ftp/arxiv/papers/1712/1712.04674.pdf
Does have it sense to use here probabilistic tools such as central limit theorem or law of iterated logarithm?
Thanks in advance for answers.
probability-theory asymptotics random-walk riemann-hypothesis
asked Jan 6 at 14:31
mkultramkultra
466
$endgroup$
add a comment |
$begingroup$
I know that the following statement:
For every $epsilon>0$
$$M(N)=O(N^{0.5+epsilon})$$ is equivalent to Riemann Hypothesis (Where $M(N)$ is Mertens function).
As Mertens function behaves somehow randomly here is idea to treat this function as a translation in random walk. Each step is related to values of independent random variable which is equal to
$$-1,0,+1$$
with probability respectively equal to
$$frac{3}{pi^{2}},1-frac{6}{pi^{2}},frac{3}{pi^{2}}$$
I know that the behaviour of translation will be more generally but on the other hand i saw a statement that given values of probability"uniquely determine the asymptotic behavior of the Mertens function"
I quoted it from this:
https://arxiv.org/ftp/arxiv/papers/1712/1712.04674.pdf
Does have it sense to use here probabilistic tools such as central limit theorem or law of iterated logarithm?
Thanks in advance for answers.
probability-theory asymptotics random-walk riemann-hypothesis
asked Jan 6 at 14:31
mkultramkultra
466
$endgroup$
I know that the following statement:
For every $epsilon>0$
$$M(N)=O(N^{0.5+epsilon})$$ is equivalent to Riemann Hypothesis (Where $M(N)$ is Mertens function).
As Mertens function behaves somehow randomly here is idea to treat this function as a translation in random walk. Each step is related to values of independent random variable which is equal to
$$-1,0,+1$$
with probability respectively equal to
$$frac{3}{pi^{2}},1-frac{6}{pi^{2}},frac{3}{pi^{2}}$$
I know that the behaviour of translation will be more generally but on the other hand i saw a statement that given values of probability"uniquely determine the asymptotic behavior of the Mertens function"
I quoted it from this:
https://arxiv.org/ftp/arxiv/papers/1712/1712.04674.pdf
Does have it sense to use here probabilistic tools such as central limit theorem or law of iterated logarithm?
Thanks in advance for answers.
probability-theory asymptotics random-walk riemann-hypothesis
probability-theory asymptotics random-walk riemann-hypothesis
asked Jan 6 at 14:31
mkultramkultra
466
asked Jan 6 at 14:31
mkultramkultra
466
asked Jan 6 at 14:31
mkultramkultra
466
asked Jan 6 at 14:31
mkultramkultra
466
asked Jan 6 at 14:31
mkultramkultra
466
466
add a comment |
add a comment |
0
active
oldest
votes
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%2f3063924%2fa-probabilistic-attempt-to-solve-riemann-hypothesis-using-mertens-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3063924%2fa-probabilistic-attempt-to-solve-riemann-hypothesis-using-mertens-function%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
