Are $lim_{x to infty} frac{Psi(x)}{pi(x)}=1$ and $pi(x)lePsi(x)$ true statements?Show...
What are the characteristics of a glide in English?
Should I use HTTPS on a domain that will only be used for redirection?
Did Amazon pay $0 in taxes last year?
If sound is a longitudinal wave, why can we hear it if our ears aren't aligned with the propagation direction?
What is better: yes / no radio, or simple checkbox?
What do you call someone who likes to pick fights?
What is the oldest European royal house?
How to increase accuracy of Plot
PTIJ: Sport in the Torah
How spaceships determine each other's mass in space?
Either of .... (Plural/Singular)
School performs periodic password audits. Is my password compromised?
In Diabelli's "Duet in D" for piano, what are these brackets on chords that look like vertical slurs?
A running toilet that stops itself
The (Easy) Road to Code
Is there a way to find out the age of climbing ropes?
Vector-transposing function
What is Tony Stark injecting into himself in Iron Man 3?
Who has more? Ireland or Iceland?
Computation logic of Partway in TikZ
Is it appropriate to ask a former professor to order a book for me through an inter-library loan?
Create chunks from an array
How would an energy-based "projectile" blow up a spaceship?
Does the US political system, in principle, allow for a no-party system?
Are $lim_{x to infty} frac{Psi(x)}{pi(x)}=1$ and $pi(x)lePsi(x)$ true statements?
Show $lim_{xto0}frac{Gamma(x)}{psi(x)}=-1$Prime Counting: Relationship between Chebyshev's function and the Prime counting functionWhat happens with $lim_{n to infty}left(frac{ln(p_n!)}{psi(p_n)}right)_n$?$lim_{ntoinfty}left(frac{log(p_{n+1})}{log(p_n)}right)^n = C?$Linear convex combinations of $Li(x)=int_2^xfrac{1}{log(t)}dt$ and $frac{x}{log(x)}$, and prime counting functionProving $lim_{n to infty} frac {log{p_n}} {log n} = 1$Why is $lim_{xtoinfty}frac{psi(x)}{x}=1$ equivalent to the prime number theorem?What's about the convergence of $sum_{n=1}^inftyfrac{mu(n)pi_2(n)}{n^2}$, on assumption of the Twin Prime Conjecture?Questions on Fourier Transform of $frac{psi[e^u]-e^u}{e^{u(1/2+epsilon)}}$Brainstorming to search links between $pi(x)$ and the convexity of $-int_2^xfrac{dt}{log(t)}$, related to the Second Hardy–Littlewood conjecture
$begingroup$
$$Psi(x)= int_2^x sin(frac{1}{2ln(t)})+sinh(frac{1}{2ln(t)})mathscr{dt}. $$
Does$$ lim_{x to infty} frac{Psi(x)}{pi(x)}=1? $$
Where $pi(x)$ is the prime counting function.
Since $$ Psi(x)approx mathscr{li}(x), $$
I believe the conjecture to be true.
Is $$pi(x)lePsi(x)?$$
number-theory limits prime-numbers asymptotics approximation
edited yesterday
rash
20811
asked yesterday
UltradarkUltradark
2551517
$endgroup$
add a comment |
$begingroup$
$$Psi(x)= int_2^x sin(frac{1}{2ln(t)})+sinh(frac{1}{2ln(t)})mathscr{dt}. $$
Does$$ lim_{x to infty} frac{Psi(x)}{pi(x)}=1? $$
Where $pi(x)$ is the prime counting function.
Since $$ Psi(x)approx mathscr{li}(x), $$
I believe the conjecture to be true.
Is $$pi(x)lePsi(x)?$$
number-theory limits prime-numbers asymptotics approximation
edited yesterday
rash
20811
asked yesterday
UltradarkUltradark
2551517
$endgroup$
1
$begingroup$
Possibly an interesting page for you: en.wikipedia.org/wiki/Skewes%27s_number
$endgroup$
– Minus One-Twelfth
yesterday
$begingroup$
n the article referenced in the comment above, it gives a reference for reent (2o15) ewsukt that pi(x)le li(x) for x<10^{19}.
$endgroup$
– DanielWainfleet
yesterday
$begingroup$
so $Psi(x)$ might behave like $mathscr{li(x)}?$
$endgroup$
– Ultradark
yesterday
$begingroup$
can we also use that: $$pi(x)approxfrac{x}{ln(x)}$$
$endgroup$
– Henry Lee
yesterday
add a comment |
$begingroup$
$$Psi(x)= int_2^x sin(frac{1}{2ln(t)})+sinh(frac{1}{2ln(t)})mathscr{dt}. $$
Does$$ lim_{x to infty} frac{Psi(x)}{pi(x)}=1? $$
Where $pi(x)$ is the prime counting function.
Since $$ Psi(x)approx mathscr{li}(x), $$
I believe the conjecture to be true.
Is $$pi(x)lePsi(x)?$$
number-theory limits prime-numbers asymptotics approximation
edited yesterday
rash
20811
asked yesterday
UltradarkUltradark
2551517
$endgroup$
$$Psi(x)= int_2^x sin(frac{1}{2ln(t)})+sinh(frac{1}{2ln(t)})mathscr{dt}. $$
Does$$ lim_{x to infty} frac{Psi(x)}{pi(x)}=1? $$
Where $pi(x)$ is the prime counting function.
Since $$ Psi(x)approx mathscr{li}(x), $$
I believe the conjecture to be true.
Is $$pi(x)lePsi(x)?$$
number-theory limits prime-numbers asymptotics approximation
number-theory limits prime-numbers asymptotics approximation
edited yesterday
rash
20811
asked yesterday
UltradarkUltradark
2551517
edited yesterday
rash
20811
asked yesterday
UltradarkUltradark
2551517
edited yesterday
rash
20811
edited yesterday
rash
20811
edited yesterday
rash
20811
20811
asked yesterday
UltradarkUltradark
2551517
asked yesterday
UltradarkUltradark
2551517
asked yesterday
UltradarkUltradark
2551517
2551517
1
$begingroup$
Possibly an interesting page for you: en.wikipedia.org/wiki/Skewes%27s_number
$endgroup$
– Minus One-Twelfth
yesterday
$begingroup$
n the article referenced in the comment above, it gives a reference for reent (2o15) ewsukt that pi(x)le li(x) for x<10^{19}.
$endgroup$
– DanielWainfleet
yesterday
$begingroup$
so $Psi(x)$ might behave like $mathscr{li(x)}?$
$endgroup$
– Ultradark
yesterday
$begingroup$
can we also use that: $$pi(x)approxfrac{x}{ln(x)}$$
$endgroup$
– Henry Lee
yesterday
add a comment |
1
$begingroup$
Possibly an interesting page for you: en.wikipedia.org/wiki/Skewes%27s_number
$endgroup$
– Minus One-Twelfth
yesterday
$begingroup$
n the article referenced in the comment above, it gives a reference for reent (2o15) ewsukt that pi(x)le li(x) for x<10^{19}.
$endgroup$
– DanielWainfleet
yesterday
$begingroup$
so $Psi(x)$ might behave like $mathscr{li(x)}?$
$endgroup$
– Ultradark
yesterday
$begingroup$
can we also use that: $$pi(x)approxfrac{x}{ln(x)}$$
$endgroup$
– Henry Lee
yesterday
1
1
$begingroup$
Possibly an interesting page for you: en.wikipedia.org/wiki/Skewes%27s_number
$endgroup$
– Minus One-Twelfth
yesterday
$begingroup$
Possibly an interesting page for you: en.wikipedia.org/wiki/Skewes%27s_number
$endgroup$
– Minus One-Twelfth
yesterday
$begingroup$
n the article referenced in the comment above, it gives a reference for reent (2o15) ewsukt that pi(x)le li(x) for x<10^{19}.
$endgroup$
– DanielWainfleet
yesterday
$begingroup$
n the article referenced in the comment above, it gives a reference for reent (2o15) ewsukt that pi(x)le li(x) for x<10^{19}.
$endgroup$
– DanielWainfleet
yesterday
$begingroup$
so $Psi(x)$ might behave like $mathscr{li(x)}?$
$endgroup$
– Ultradark
yesterday
$begingroup$
so $Psi(x)$ might behave like $mathscr{li(x)}?$
$endgroup$
– Ultradark
yesterday
$begingroup$
can we also use that: $$pi(x)approxfrac{x}{ln(x)}$$
$endgroup$
– Henry Lee
yesterday
$begingroup$
can we also use that: $$pi(x)approxfrac{x}{ln(x)}$$
$endgroup$
– Henry Lee
yesterday
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
If it is true that $Psi(x)approx text{Li}(x)$ and we use that $pi(x)approxfrac{x}{ln(x)}$ then we can say that:
$$lim_{xtoinfty}frac{Psi(x)}{pi(x)}=lim_{xtoinfty}frac{ln(x)int_0^xfrac{dt}{ln(t)}}{x}$$
And now use L'Hopitals rule. Failing this use Squeeze theorem?
answered yesterday
Henry LeeHenry Lee
2,129219
$endgroup$
add a comment |
$begingroup$
Since $sin x=x+o(x)$ and $sinh x=x+o(x)$,
$$Psi (x)=int^x_2 frac1{ln t}dt+int^x_2 o(ln t)dt=operatorname{Li}(x)+o(operatorname{Li}(x))$$
Hence,
$$frac{Psi(x)}{pi(x)}simfrac{operatorname{Li}(x)}{pi(x)}sim 1$$
provided that $operatorname{Li}(x)/pi(x)to1$.
answered yesterday
SzetoSzeto
6,6512926
$endgroup$
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%2f3139524%2fare-lim-x-to-infty-frac-psix-pix-1-and-pix-le-psix-true-s%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If it is true that $Psi(x)approx text{Li}(x)$ and we use that $pi(x)approxfrac{x}{ln(x)}$ then we can say that:
$$lim_{xtoinfty}frac{Psi(x)}{pi(x)}=lim_{xtoinfty}frac{ln(x)int_0^xfrac{dt}{ln(t)}}{x}$$
And now use L'Hopitals rule. Failing this use Squeeze theorem?
answered yesterday
Henry LeeHenry Lee
2,129219
$endgroup$
add a comment |
$begingroup$
If it is true that $Psi(x)approx text{Li}(x)$ and we use that $pi(x)approxfrac{x}{ln(x)}$ then we can say that:
$$lim_{xtoinfty}frac{Psi(x)}{pi(x)}=lim_{xtoinfty}frac{ln(x)int_0^xfrac{dt}{ln(t)}}{x}$$
And now use L'Hopitals rule. Failing this use Squeeze theorem?
answered yesterday
Henry LeeHenry Lee
2,129219
$endgroup$
add a comment |
$begingroup$
If it is true that $Psi(x)approx text{Li}(x)$ and we use that $pi(x)approxfrac{x}{ln(x)}$ then we can say that:
$$lim_{xtoinfty}frac{Psi(x)}{pi(x)}=lim_{xtoinfty}frac{ln(x)int_0^xfrac{dt}{ln(t)}}{x}$$
And now use L'Hopitals rule. Failing this use Squeeze theorem?
answered yesterday
Henry LeeHenry Lee
2,129219
$endgroup$
If it is true that $Psi(x)approx text{Li}(x)$ and we use that $pi(x)approxfrac{x}{ln(x)}$ then we can say that:
$$lim_{xtoinfty}frac{Psi(x)}{pi(x)}=lim_{xtoinfty}frac{ln(x)int_0^xfrac{dt}{ln(t)}}{x}$$
And now use L'Hopitals rule. Failing this use Squeeze theorem?
answered yesterday
Henry LeeHenry Lee
2,129219
answered yesterday
Henry LeeHenry Lee
2,129219
answered yesterday
Henry LeeHenry Lee
2,129219
answered yesterday
Henry LeeHenry Lee
2,129219
2,129219
add a comment |
add a comment |
$begingroup$
Since $sin x=x+o(x)$ and $sinh x=x+o(x)$,
$$Psi (x)=int^x_2 frac1{ln t}dt+int^x_2 o(ln t)dt=operatorname{Li}(x)+o(operatorname{Li}(x))$$
Hence,
$$frac{Psi(x)}{pi(x)}simfrac{operatorname{Li}(x)}{pi(x)}sim 1$$
provided that $operatorname{Li}(x)/pi(x)to1$.
answered yesterday
SzetoSzeto
6,6512926
$endgroup$
add a comment |
$begingroup$
Since $sin x=x+o(x)$ and $sinh x=x+o(x)$,
$$Psi (x)=int^x_2 frac1{ln t}dt+int^x_2 o(ln t)dt=operatorname{Li}(x)+o(operatorname{Li}(x))$$
Hence,
$$frac{Psi(x)}{pi(x)}simfrac{operatorname{Li}(x)}{pi(x)}sim 1$$
provided that $operatorname{Li}(x)/pi(x)to1$.
answered yesterday
SzetoSzeto
6,6512926
$endgroup$
add a comment |
$begingroup$
Since $sin x=x+o(x)$ and $sinh x=x+o(x)$,
$$Psi (x)=int^x_2 frac1{ln t}dt+int^x_2 o(ln t)dt=operatorname{Li}(x)+o(operatorname{Li}(x))$$
Hence,
$$frac{Psi(x)}{pi(x)}simfrac{operatorname{Li}(x)}{pi(x)}sim 1$$
provided that $operatorname{Li}(x)/pi(x)to1$.
answered yesterday
SzetoSzeto
6,6512926
$endgroup$
Since $sin x=x+o(x)$ and $sinh x=x+o(x)$,
$$Psi (x)=int^x_2 frac1{ln t}dt+int^x_2 o(ln t)dt=operatorname{Li}(x)+o(operatorname{Li}(x))$$
Hence,
$$frac{Psi(x)}{pi(x)}simfrac{operatorname{Li}(x)}{pi(x)}sim 1$$
provided that $operatorname{Li}(x)/pi(x)to1$.
answered yesterday
SzetoSzeto
6,6512926
edited yesterday
answered yesterday
SzetoSzeto
6,6512926
answered yesterday
SzetoSzeto
6,6512926
answered yesterday
SzetoSzeto
6,6512926
6,6512926
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%2f3139524%2fare-lim-x-to-infty-frac-psix-pix-1-and-pix-le-psix-true-s%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
1
$begingroup$
Possibly an interesting page for you: en.wikipedia.org/wiki/Skewes%27s_number
$endgroup$
– Minus One-Twelfth
yesterday
$begingroup$
n the article referenced in the comment above, it gives a reference for reent (2o15) ewsukt that pi(x)le li(x) for x<10^{19}.
$endgroup$
– DanielWainfleet
yesterday
$begingroup$
so $Psi(x)$ might behave like $mathscr{li(x)}?$
$endgroup$
– Ultradark
yesterday
$begingroup$
can we also use that: $$pi(x)approxfrac{x}{ln(x)}$$
$endgroup$
– Henry Lee
yesterday