Relation between The Euler Totient, the counting prime formula and the prime generating Functions [closed]
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](https://i.stack.imgur.com/lKFS0.png)
Relation between
The Euler Totient,
the counting prime formula
and the prime generating Functions
There is a formula for the ivisor sum hiih is one of the most useful
propertes of the Euler Formula to fond a relations with prime counting Formula ans sum of primes.
Proof : http://vixra.org/abs/1901.0046
number-theory prime-numbers prime-factorization divisor-counting-function
asked Jan 5 at 14:44
Na ZihNa Zih
11
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closed as unclear what you're asking by Namaste, Gottfried Helms, user416281, pre-kidney, metamorphy Jan 5 at 18:40
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
](https://i.stack.imgur.com/lKFS0.png)
Relation between
The Euler Totient,
the counting prime formula
and the prime generating Functions
There is a formula for the ivisor sum hiih is one of the most useful
propertes of the Euler Formula to fond a relations with prime counting Formula ans sum of primes.
Proof : http://vixra.org/abs/1901.0046
number-theory prime-numbers prime-factorization divisor-counting-function
asked Jan 5 at 14:44
Na ZihNa Zih
11
$endgroup$
closed as unclear what you're asking by Namaste, Gottfried Helms, user416281, pre-kidney, metamorphy Jan 5 at 18:40
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
](https://i.stack.imgur.com/lKFS0.png)
Relation between
The Euler Totient,
the counting prime formula
and the prime generating Functions
There is a formula for the ivisor sum hiih is one of the most useful
propertes of the Euler Formula to fond a relations with prime counting Formula ans sum of primes.
Proof : http://vixra.org/abs/1901.0046
number-theory prime-numbers prime-factorization divisor-counting-function
asked Jan 5 at 14:44
Na ZihNa Zih
11
$endgroup$
](https://i.stack.imgur.com/lKFS0.png)
Relation between
The Euler Totient,
the counting prime formula
and the prime generating Functions
There is a formula for the ivisor sum hiih is one of the most useful
propertes of the Euler Formula to fond a relations with prime counting Formula ans sum of primes.
Proof : http://vixra.org/abs/1901.0046
number-theory prime-numbers prime-factorization divisor-counting-function
number-theory prime-numbers prime-factorization divisor-counting-function
asked Jan 5 at 14:44
Na ZihNa Zih
11
asked Jan 5 at 14:44
Na ZihNa Zih
11
asked Jan 5 at 14:44
Na ZihNa Zih
11
asked Jan 5 at 14:44
Na ZihNa Zih
11
asked Jan 5 at 14:44
Na ZihNa Zih
11
11
closed as unclear what you're asking by Namaste, Gottfried Helms, user416281, pre-kidney, metamorphy Jan 5 at 18:40
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Namaste, Gottfried Helms, user416281, pre-kidney, metamorphy Jan 5 at 18:40
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
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1 Answer
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You are not really linking $varphi(n)$ and $pi(n)$. You are just adding things that cancel out.
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid ,n}1+sumlimits_{dneq pmid n}varphi(d)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{dmid n}varphi(d)-sumlimits_{pmid n}varphi(p)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+n-sumlimits_{pmid n}(p-1)-n$$
$$pi(n)=sumlimits_{pmid n}p-sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{pmid n}1+n-n$$
$$pi(n)=sumlimits_{pleq n}1$$
$pi(n)$ is hidden in the second term
$$sumlimits_{pleq n, p,nmid, n}1$$
You could do that with any formula. Was there any reason to split it that way?
edited Jan 5 at 17:57
reuns
20.5k21352
answered Jan 5 at 16:22
Collag3nCollag3n
784211
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You are not really linking $varphi(n)$ and $pi(n)$. You are just adding things that cancel out.
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid ,n}1+sumlimits_{dneq pmid n}varphi(d)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{dmid n}varphi(d)-sumlimits_{pmid n}varphi(p)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+n-sumlimits_{pmid n}(p-1)-n$$
$$pi(n)=sumlimits_{pmid n}p-sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{pmid n}1+n-n$$
$$pi(n)=sumlimits_{pleq n}1$$
$pi(n)$ is hidden in the second term
$$sumlimits_{pleq n, p,nmid, n}1$$
You could do that with any formula. Was there any reason to split it that way?
edited Jan 5 at 17:57
reuns
20.5k21352
answered Jan 5 at 16:22
Collag3nCollag3n
784211
$endgroup$
add a comment |
$begingroup$
You are not really linking $varphi(n)$ and $pi(n)$. You are just adding things that cancel out.
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid ,n}1+sumlimits_{dneq pmid n}varphi(d)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{dmid n}varphi(d)-sumlimits_{pmid n}varphi(p)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+n-sumlimits_{pmid n}(p-1)-n$$
$$pi(n)=sumlimits_{pmid n}p-sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{pmid n}1+n-n$$
$$pi(n)=sumlimits_{pleq n}1$$
$pi(n)$ is hidden in the second term
$$sumlimits_{pleq n, p,nmid, n}1$$
You could do that with any formula. Was there any reason to split it that way?
edited Jan 5 at 17:57
reuns
20.5k21352
answered Jan 5 at 16:22
Collag3nCollag3n
784211
$endgroup$
add a comment |
$begingroup$
You are not really linking $varphi(n)$ and $pi(n)$. You are just adding things that cancel out.
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid ,n}1+sumlimits_{dneq pmid n}varphi(d)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{dmid n}varphi(d)-sumlimits_{pmid n}varphi(p)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+n-sumlimits_{pmid n}(p-1)-n$$
$$pi(n)=sumlimits_{pmid n}p-sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{pmid n}1+n-n$$
$$pi(n)=sumlimits_{pleq n}1$$
$pi(n)$ is hidden in the second term
$$sumlimits_{pleq n, p,nmid, n}1$$
You could do that with any formula. Was there any reason to split it that way?
edited Jan 5 at 17:57
reuns
20.5k21352
answered Jan 5 at 16:22
Collag3nCollag3n
784211
$endgroup$
You are not really linking $varphi(n)$ and $pi(n)$. You are just adding things that cancel out.
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid ,n}1+sumlimits_{dneq pmid n}varphi(d)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{dmid n}varphi(d)-sumlimits_{pmid n}varphi(p)-n$$
$$pi(n)=sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+n-sumlimits_{pmid n}(p-1)-n$$
$$pi(n)=sumlimits_{pmid n}p-sumlimits_{pmid n}p+sumlimits_{pleq n, p,nmid, n}1+sumlimits_{pmid n}1+n-n$$
$$pi(n)=sumlimits_{pleq n}1$$
$pi(n)$ is hidden in the second term
$$sumlimits_{pleq n, p,nmid, n}1$$
You could do that with any formula. Was there any reason to split it that way?
edited Jan 5 at 17:57
reuns
20.5k21352
answered Jan 5 at 16:22
Collag3nCollag3n
784211
edited Jan 5 at 17:57
reuns
20.5k21352
edited Jan 5 at 17:57
reuns
20.5k21352
edited Jan 5 at 17:57
reuns
20.5k21352
20.5k21352
answered Jan 5 at 16:22
Collag3nCollag3n
784211
answered Jan 5 at 16:22
Collag3nCollag3n
784211
answered Jan 5 at 16:22
Collag3nCollag3n
784211
784211
add a comment |
add a comment |
