Sum of the number of relatively prime integers up to $x$, $x-1$, $ldots$, $1$Counting elements of reduced residue systems modulo one number which are smaller than anotherEuler's totient function Proof that $nmid varphi(2^n-1)$Euler's totient function for large numbersNumber theory question about Euler's totient functionDoes the totient function reach all its values when restricted to odd numbers?Find all values of N that satisfy ϕ(N) = x for any given x valueOn factoring and integer given the value of its Euler's totient function.How to count the number of perfect square greater than $N$ and less than $N^2$ that are relatively prime to $N$?How many numbers less than $m$ and relatively prime to $n$, where $m>n$?Showing that two binomials are relatively prime for all positive integers (Euclidean ?)
Irreducibility of a simple polynomial
Opposite of a diet
Failed to fetch jessie backports repository
Are there any comparative studies done between Ashtavakra Gita and Buddhim?
Minimal reference content
Coordinate position not precise
What would happen if the UK refused to take part in EU Parliamentary elections?
How can a jailer prevent the Forge Cleric's Artisan's Blessing from being used?
What's a natural way to say that someone works somewhere (for a job)?
Mapping a list into a phase plot
Can I use my Chinese passport to enter China after I acquired another citizenship?
Is exact Kanji stroke length important?
Why did Kant, Hegel, and Adorno leave some words and phrases in the Greek alphabet?
How do I rename a LINUX host without needing to reboot for the rename to take effect?
Can a monster with multiattack use this ability if they are missing a limb?
Lay out the Carpet
Products and sum of cubes in Fibonacci
Have I saved too much for retirement so far?
What is the oldest known work of fiction?
Where in the Bible does the greeting ("Dominus Vobiscum") used at Mass come from?
Is it correct to write "is not focus on"?
Valid Badminton Score?
Should my PhD thesis be submitted under my legal name?
Transcription Beats per minute
Sum of the number of relatively prime integers up to $x$, $x-1$, $ldots$, $1$
Counting elements of reduced residue systems modulo one number which are smaller than anotherEuler's totient function Proof that $nmid varphi(2^n-1)$Euler's totient function for large numbersNumber theory question about Euler's totient functionDoes the totient function reach all its values when restricted to odd numbers?Find all values of N that satisfy ϕ(N) = x for any given x valueOn factoring and integer given the value of its Euler's totient function.How to count the number of perfect square greater than $N$ and less than $N^2$ that are relatively prime to $N$?How many numbers less than $m$ and relatively prime to $n$, where $m>n$?Showing that two binomials are relatively prime for all positive integers (Euclidean ?)
$begingroup$
If there is a number $x$, and we want to find the sum of the number of relatively prime integers up to $x$, $x-1$, $dots$ until $1$, is there a formula for this or any way to solve it? Like if the number is $6$, you add up $0$ (for numbers relatively prime to $1$), $1$ (for numbers relatively prime to $2$), $2$ (for $3$), $2$ (for $4$), $4$ (for $5$), and $2$ (for $6$) for a total of $11$ numbers. I tried using Euler's Totient Function, but with a high number, that would require far too many computations. Is there any way to compute this for a high number, let's say $2019$, without taking a lot of time?
totient-function coprime
edited Mar 17 at 11:19
Rócherz
3,0013821
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
$endgroup$
|
show 3 more comments
$begingroup$
If there is a number $x$, and we want to find the sum of the number of relatively prime integers up to $x$, $x-1$, $dots$ until $1$, is there a formula for this or any way to solve it? Like if the number is $6$, you add up $0$ (for numbers relatively prime to $1$), $1$ (for numbers relatively prime to $2$), $2$ (for $3$), $2$ (for $4$), $4$ (for $5$), and $2$ (for $6$) for a total of $11$ numbers. I tried using Euler's Totient Function, but with a high number, that would require far too many computations. Is there any way to compute this for a high number, let's say $2019$, without taking a lot of time?
totient-function coprime
edited Mar 17 at 11:19
Rócherz
3,0013821
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
$endgroup$
1
$begingroup$
mathworld.wolfram.com/TotientSummatoryFunction.html
$endgroup$
– saulspatz
Mar 15 at 23:30
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 15 at 23:46
$begingroup$
So basically you want to compute $sum_ileq n varphi(i)$?
$endgroup$
– tomasz
Mar 15 at 23:50
$begingroup$
Yes but how do you compute it?
$endgroup$
– Jaemin Kim
Mar 16 at 0:17
$begingroup$
"let's say 2019" That's oddly specific. Where is this problem from, exactly? And are you supposed to calculate it by hand?
$endgroup$
– Arthur
Mar 16 at 0:50
|
show 3 more comments
$begingroup$
If there is a number $x$, and we want to find the sum of the number of relatively prime integers up to $x$, $x-1$, $dots$ until $1$, is there a formula for this or any way to solve it? Like if the number is $6$, you add up $0$ (for numbers relatively prime to $1$), $1$ (for numbers relatively prime to $2$), $2$ (for $3$), $2$ (for $4$), $4$ (for $5$), and $2$ (for $6$) for a total of $11$ numbers. I tried using Euler's Totient Function, but with a high number, that would require far too many computations. Is there any way to compute this for a high number, let's say $2019$, without taking a lot of time?
totient-function coprime
edited Mar 17 at 11:19
Rócherz
3,0013821
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
$endgroup$
If there is a number $x$, and we want to find the sum of the number of relatively prime integers up to $x$, $x-1$, $dots$ until $1$, is there a formula for this or any way to solve it? Like if the number is $6$, you add up $0$ (for numbers relatively prime to $1$), $1$ (for numbers relatively prime to $2$), $2$ (for $3$), $2$ (for $4$), $4$ (for $5$), and $2$ (for $6$) for a total of $11$ numbers. I tried using Euler's Totient Function, but with a high number, that would require far too many computations. Is there any way to compute this for a high number, let's say $2019$, without taking a lot of time?
totient-function coprime
totient-function coprime
edited Mar 17 at 11:19
Rócherz
3,0013821
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
edited Mar 17 at 11:19
Rócherz
3,0013821
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
edited Mar 17 at 11:19
Rócherz
3,0013821
edited Mar 17 at 11:19
Rócherz
3,0013821
edited Mar 17 at 11:19
Rócherz
3,0013821
3,0013821
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
asked Mar 15 at 23:25
Jaemin KimJaemin Kim
31
31
1
$begingroup$
mathworld.wolfram.com/TotientSummatoryFunction.html
$endgroup$
– saulspatz
Mar 15 at 23:30
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 15 at 23:46
$begingroup$
So basically you want to compute $sum_ileq n varphi(i)$?
$endgroup$
– tomasz
Mar 15 at 23:50
$begingroup$
Yes but how do you compute it?
$endgroup$
– Jaemin Kim
Mar 16 at 0:17
$begingroup$
"let's say 2019" That's oddly specific. Where is this problem from, exactly? And are you supposed to calculate it by hand?
$endgroup$
– Arthur
Mar 16 at 0:50
|
show 3 more comments
1
$begingroup$
mathworld.wolfram.com/TotientSummatoryFunction.html
$endgroup$
– saulspatz
Mar 15 at 23:30
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 15 at 23:46
$begingroup$
So basically you want to compute $sum_ileq n varphi(i)$?
$endgroup$
– tomasz
Mar 15 at 23:50
$begingroup$
Yes but how do you compute it?
$endgroup$
– Jaemin Kim
Mar 16 at 0:17
$begingroup$
"let's say 2019" That's oddly specific. Where is this problem from, exactly? And are you supposed to calculate it by hand?
$endgroup$
– Arthur
Mar 16 at 0:50
1
1
$begingroup$
mathworld.wolfram.com/TotientSummatoryFunction.html
$endgroup$
– saulspatz
Mar 15 at 23:30
$begingroup$
mathworld.wolfram.com/TotientSummatoryFunction.html
$endgroup$
– saulspatz
Mar 15 at 23:30
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 15 at 23:46
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 15 at 23:46
$begingroup$
So basically you want to compute $sum_ileq n varphi(i)$?
$endgroup$
– tomasz
Mar 15 at 23:50
$begingroup$
So basically you want to compute $sum_ileq n varphi(i)$?
$endgroup$
– tomasz
Mar 15 at 23:50
$begingroup$
Yes but how do you compute it?
$endgroup$
– Jaemin Kim
Mar 16 at 0:17
$begingroup$
Yes but how do you compute it?
$endgroup$
– Jaemin Kim
Mar 16 at 0:17
$begingroup$
"let's say 2019" That's oddly specific. Where is this problem from, exactly? And are you supposed to calculate it by hand?
$endgroup$
– Arthur
Mar 16 at 0:50
$begingroup$
"let's say 2019" That's oddly specific. Where is this problem from, exactly? And are you supposed to calculate it by hand?
$endgroup$
– Arthur
Mar 16 at 0:50
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
There is a well-known expression for the Euler function using the Möbius function:
$$ varphi(n) = nsum_dmid nfracmu(d)d. $$
Consequently,
beginalign
sum_n=1^N varphi(n)
&= sum_n=1^N n sum_dmid n fracmu(d)d \
&= sum_d=1^N fracmu(d)d sum_kle N/d kd \
&= frac12,sum_d=1^N mu(d) leftlfloor frac Ndright rfloor left(leftlfloor frac Ndright rfloor +1right).
endalign
This expression is standartly used to give an asymptotic for the sum in the LHS, but it also can be used to efficiently calculate this sum.
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
$endgroup$
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%2f3149907%2fsum-of-the-number-of-relatively-prime-integers-up-to-x-x-1-ldots-1%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
$begingroup$
There is a well-known expression for the Euler function using the Möbius function:
$$ varphi(n) = nsum_dmid nfracmu(d)d. $$
Consequently,
beginalign
sum_n=1^N varphi(n)
&= sum_n=1^N n sum_dmid n fracmu(d)d \
&= sum_d=1^N fracmu(d)d sum_kle N/d kd \
&= frac12,sum_d=1^N mu(d) leftlfloor frac Ndright rfloor left(leftlfloor frac Ndright rfloor +1right).
endalign
This expression is standartly used to give an asymptotic for the sum in the LHS, but it also can be used to efficiently calculate this sum.
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
$endgroup$
add a comment |
$begingroup$
There is a well-known expression for the Euler function using the Möbius function:
$$ varphi(n) = nsum_dmid nfracmu(d)d. $$
Consequently,
beginalign
sum_n=1^N varphi(n)
&= sum_n=1^N n sum_dmid n fracmu(d)d \
&= sum_d=1^N fracmu(d)d sum_kle N/d kd \
&= frac12,sum_d=1^N mu(d) leftlfloor frac Ndright rfloor left(leftlfloor frac Ndright rfloor +1right).
endalign
This expression is standartly used to give an asymptotic for the sum in the LHS, but it also can be used to efficiently calculate this sum.
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
$endgroup$
add a comment |
$begingroup$
There is a well-known expression for the Euler function using the Möbius function:
$$ varphi(n) = nsum_dmid nfracmu(d)d. $$
Consequently,
beginalign
sum_n=1^N varphi(n)
&= sum_n=1^N n sum_dmid n fracmu(d)d \
&= sum_d=1^N fracmu(d)d sum_kle N/d kd \
&= frac12,sum_d=1^N mu(d) leftlfloor frac Ndright rfloor left(leftlfloor frac Ndright rfloor +1right).
endalign
This expression is standartly used to give an asymptotic for the sum in the LHS, but it also can be used to efficiently calculate this sum.
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
$endgroup$
There is a well-known expression for the Euler function using the Möbius function:
$$ varphi(n) = nsum_dmid nfracmu(d)d. $$
Consequently,
beginalign
sum_n=1^N varphi(n)
&= sum_n=1^N n sum_dmid n fracmu(d)d \
&= sum_d=1^N fracmu(d)d sum_kle N/d kd \
&= frac12,sum_d=1^N mu(d) leftlfloor frac Ndright rfloor left(leftlfloor frac Ndright rfloor +1right).
endalign
This expression is standartly used to give an asymptotic for the sum in the LHS, but it also can be used to efficiently calculate this sum.
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
answered Mar 17 at 11:37
W-t-PW-t-P
1,529612
1,529612
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%2f3149907%2fsum-of-the-number-of-relatively-prime-integers-up-to-x-x-1-ldots-1%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
$begingroup$
mathworld.wolfram.com/TotientSummatoryFunction.html
$endgroup$
– saulspatz
Mar 15 at 23:30
$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax.
$endgroup$
– dantopa
Mar 15 at 23:46
$begingroup$
So basically you want to compute $sum_ileq n varphi(i)$?
$endgroup$
– tomasz
Mar 15 at 23:50
$begingroup$
Yes but how do you compute it?
$endgroup$
– Jaemin Kim
Mar 16 at 0:17
$begingroup$
"let's say 2019" That's oddly specific. Where is this problem from, exactly? And are you supposed to calculate it by hand?
$endgroup$
– Arthur
Mar 16 at 0:50