Mixing
Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?
Clash Royale CLAN TAG#URR8PPP
up vote
10
down vote
favorite
I was looking at a list of primes. I noticed that $ fracAM (p_1, p_2, ldots, p_n)p_n$ seemed to converge.
This led me to try $ fracGM (p_1, p_2, ldots, p_n)p_n$ which also seemed to converge.
I did a quick Excel graph and regression and found the former seemed to converge to $frac12$ and latter to $frac1e$. As with anything related to primes, no easy reasoning seemed to point to those results (however, for all natural numbers it was trivial to show that the former asymptotically tended to $frac12$).
Are these observations correct and are there any proofs towards:
$$
lim_ntoinfty left( fracAM (p_1, p_2, ldots, p_n)p_n right)
= frac12 tag1
$$
$$
lim_ntoinfty left( fracGM (p_1, p_2, ldots, p_n)p_n right)
= frac1e tag2
$$
number-theory elementary-number-theory prime-numbers
edited 36 mins ago
Théophile
17.2k12438
asked 44 mins ago
Soham
1324
add a comment |Â
up vote
10
down vote
favorite
I was looking at a list of primes. I noticed that $ fracAM (p_1, p_2, ldots, p_n)p_n$ seemed to converge.
This led me to try $ fracGM (p_1, p_2, ldots, p_n)p_n$ which also seemed to converge.
I did a quick Excel graph and regression and found the former seemed to converge to $frac12$ and latter to $frac1e$. As with anything related to primes, no easy reasoning seemed to point to those results (however, for all natural numbers it was trivial to show that the former asymptotically tended to $frac12$).
Are these observations correct and are there any proofs towards:
$$
lim_ntoinfty left( fracAM (p_1, p_2, ldots, p_n)p_n right)
= frac12 tag1
$$
$$
lim_ntoinfty left( fracGM (p_1, p_2, ldots, p_n)p_n right)
= frac1e tag2
$$
number-theory elementary-number-theory prime-numbers
edited 36 mins ago
Théophile
17.2k12438
asked 44 mins ago
Soham
1324
Can we take $p_n = ncdot log(n)$ in these limits? (using the prime numbers theorem)
– Yanko
22 mins ago
add a comment |Â
up vote
10
down vote
favorite
up vote
10
down vote
favorite
I was looking at a list of primes. I noticed that $ fracAM (p_1, p_2, ldots, p_n)p_n$ seemed to converge.
This led me to try $ fracGM (p_1, p_2, ldots, p_n)p_n$ which also seemed to converge.
I did a quick Excel graph and regression and found the former seemed to converge to $frac12$ and latter to $frac1e$. As with anything related to primes, no easy reasoning seemed to point to those results (however, for all natural numbers it was trivial to show that the former asymptotically tended to $frac12$).
Are these observations correct and are there any proofs towards:
$$
lim_ntoinfty left( fracAM (p_1, p_2, ldots, p_n)p_n right)
= frac12 tag1
$$
$$
lim_ntoinfty left( fracGM (p_1, p_2, ldots, p_n)p_n right)
= frac1e tag2
$$
number-theory elementary-number-theory prime-numbers
edited 36 mins ago
Théophile
17.2k12438
asked 44 mins ago
Soham
1324
I was looking at a list of primes. I noticed that $ fracAM (p_1, p_2, ldots, p_n)p_n$ seemed to converge.
This led me to try $ fracGM (p_1, p_2, ldots, p_n)p_n$ which also seemed to converge.
I did a quick Excel graph and regression and found the former seemed to converge to $frac12$ and latter to $frac1e$. As with anything related to primes, no easy reasoning seemed to point to those results (however, for all natural numbers it was trivial to show that the former asymptotically tended to $frac12$).
Are these observations correct and are there any proofs towards:
$$
lim_ntoinfty left( fracAM (p_1, p_2, ldots, p_n)p_n right)
= frac12 tag1
$$
$$
lim_ntoinfty left( fracGM (p_1, p_2, ldots, p_n)p_n right)
= frac1e tag2
$$
number-theory elementary-number-theory prime-numbers
number-theory elementary-number-theory prime-numbers
edited 36 mins ago
Théophile
17.2k12438
asked 44 mins ago
Soham
1324
edited 36 mins ago
Théophile
17.2k12438
asked 44 mins ago
Soham
1324
edited 36 mins ago
Théophile
17.2k12438
edited 36 mins ago
Théophile
17.2k12438
edited 36 mins ago
Théophile
17.2k12438
17.2k12438
asked 44 mins ago
Soham
1324
asked 44 mins ago
Soham
1324
asked 44 mins ago
Soham
1324
1324
Can we take $p_n = ncdot log(n)$ in these limits? (using the prime numbers theorem)
– Yanko
22 mins ago
add a comment |Â
Can we take $p_n = ncdot log(n)$ in these limits? (using the prime numbers theorem)
– Yanko
22 mins ago
Can we take $p_n = ncdot log(n)$ in these limits? (using the prime numbers theorem)
– Yanko
22 mins ago
Can we take $p_n = ncdot log(n)$ in these limits? (using the prime numbers theorem)
– Yanko
22 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
We can use the simple version of the prime counting function
$$p_n approx n log n$$ and plug it into your expressions. For the arithmetic one, this becomes $$lim_n to infty frac sum_i=1^n p_inp_n=lim_n to inftyfrac log(H(n))n^2log(n)$$
Where $H(n)$ is the hyperfactorial function. We can use the expansion given on the Mathworld page to get
$$log H(n)approx log A -frac n^24+left(frac n(n+1)2+frac 112right)log (n)$$
and the limit is duly $frac 12$
I didn't find a nice expression for the product of the primes.
answered 10 mins ago
Ross Millikan
280k23189356
add a comment |Â
up vote
2
down vote
Your conjecture for GM was proved in 2011 in the short paper On a limit involving the product of prime numbers by József Sándor and Antoine Verroken.
Abstract. Let $p_k$ denote the $k$th prime number. The aim of this note is to prove that the limit of the sequence $(p_n / sqrt[n]p_1 cdots p_n)$ is $e$.
The authors obtain the result based on the prime number theorem, i.e.,
$$p approx n log n quad textrmas n to infty$$
as well as an inequality with Chebyshev's function $$theta(x) = sum_p le xlog p$$
where $p$ are primes less than $x$.
answered 12 mins ago
Théophile
17.2k12438
That contradicts my numerical observations. Could you please point out where I've gone wrong?
– TheSimpliFire
9 mins ago
@TheSimpliFire I think we simply can't conclude that the sequence is strictly decreasing based on the first $15$ values. In a paper from 2016 on the same topic, the author points out that the sequence converges slowly.
– Théophile
6 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
We can use the simple version of the prime counting function
$$p_n approx n log n$$ and plug it into your expressions. For the arithmetic one, this becomes $$lim_n to infty frac sum_i=1^n p_inp_n=lim_n to inftyfrac log(H(n))n^2log(n)$$
Where $H(n)$ is the hyperfactorial function. We can use the expansion given on the Mathworld page to get
$$log H(n)approx log A -frac n^24+left(frac n(n+1)2+frac 112right)log (n)$$
and the limit is duly $frac 12$
I didn't find a nice expression for the product of the primes.
answered 10 mins ago
Ross Millikan
280k23189356
add a comment |Â
up vote
3
down vote
We can use the simple version of the prime counting function
$$p_n approx n log n$$ and plug it into your expressions. For the arithmetic one, this becomes $$lim_n to infty frac sum_i=1^n p_inp_n=lim_n to inftyfrac log(H(n))n^2log(n)$$
Where $H(n)$ is the hyperfactorial function. We can use the expansion given on the Mathworld page to get
$$log H(n)approx log A -frac n^24+left(frac n(n+1)2+frac 112right)log (n)$$
and the limit is duly $frac 12$
I didn't find a nice expression for the product of the primes.
answered 10 mins ago
Ross Millikan
280k23189356
add a comment |Â
up vote
3
down vote
up vote
3
down vote
We can use the simple version of the prime counting function
$$p_n approx n log n$$ and plug it into your expressions. For the arithmetic one, this becomes $$lim_n to infty frac sum_i=1^n p_inp_n=lim_n to inftyfrac log(H(n))n^2log(n)$$
Where $H(n)$ is the hyperfactorial function. We can use the expansion given on the Mathworld page to get
$$log H(n)approx log A -frac n^24+left(frac n(n+1)2+frac 112right)log (n)$$
and the limit is duly $frac 12$
I didn't find a nice expression for the product of the primes.
answered 10 mins ago
Ross Millikan
280k23189356
We can use the simple version of the prime counting function
$$p_n approx n log n$$ and plug it into your expressions. For the arithmetic one, this becomes $$lim_n to infty frac sum_i=1^n p_inp_n=lim_n to inftyfrac log(H(n))n^2log(n)$$
Where $H(n)$ is the hyperfactorial function. We can use the expansion given on the Mathworld page to get
$$log H(n)approx log A -frac n^24+left(frac n(n+1)2+frac 112right)log (n)$$
and the limit is duly $frac 12$
I didn't find a nice expression for the product of the primes.
answered 10 mins ago
Ross Millikan
280k23189356
answered 10 mins ago
Ross Millikan
280k23189356
answered 10 mins ago
Ross Millikan
280k23189356
answered 10 mins ago
Ross Millikan
280k23189356
280k23189356
add a comment |Â
add a comment |Â
up vote
2
down vote
Your conjecture for GM was proved in 2011 in the short paper On a limit involving the product of prime numbers by József Sándor and Antoine Verroken.
Abstract. Let $p_k$ denote the $k$th prime number. The aim of this note is to prove that the limit of the sequence $(p_n / sqrt[n]p_1 cdots p_n)$ is $e$.
The authors obtain the result based on the prime number theorem, i.e.,
$$p approx n log n quad textrmas n to infty$$
as well as an inequality with Chebyshev's function $$theta(x) = sum_p le xlog p$$
where $p$ are primes less than $x$.
answered 12 mins ago
Théophile
17.2k12438
That contradicts my numerical observations. Could you please point out where I've gone wrong?
– TheSimpliFire
9 mins ago
@TheSimpliFire I think we simply can't conclude that the sequence is strictly decreasing based on the first $15$ values. In a paper from 2016 on the same topic, the author points out that the sequence converges slowly.
– Théophile
6 mins ago
add a comment |Â
up vote
2
down vote
Your conjecture for GM was proved in 2011 in the short paper On a limit involving the product of prime numbers by József Sándor and Antoine Verroken.
Abstract. Let $p_k$ denote the $k$th prime number. The aim of this note is to prove that the limit of the sequence $(p_n / sqrt[n]p_1 cdots p_n)$ is $e$.
The authors obtain the result based on the prime number theorem, i.e.,
$$p approx n log n quad textrmas n to infty$$
as well as an inequality with Chebyshev's function $$theta(x) = sum_p le xlog p$$
where $p$ are primes less than $x$.
answered 12 mins ago
Théophile
17.2k12438
That contradicts my numerical observations. Could you please point out where I've gone wrong?
– TheSimpliFire
9 mins ago
@TheSimpliFire I think we simply can't conclude that the sequence is strictly decreasing based on the first $15$ values. In a paper from 2016 on the same topic, the author points out that the sequence converges slowly.
– Théophile
6 mins ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Your conjecture for GM was proved in 2011 in the short paper On a limit involving the product of prime numbers by József Sándor and Antoine Verroken.
Abstract. Let $p_k$ denote the $k$th prime number. The aim of this note is to prove that the limit of the sequence $(p_n / sqrt[n]p_1 cdots p_n)$ is $e$.
The authors obtain the result based on the prime number theorem, i.e.,
$$p approx n log n quad textrmas n to infty$$
as well as an inequality with Chebyshev's function $$theta(x) = sum_p le xlog p$$
where $p$ are primes less than $x$.
answered 12 mins ago
Théophile
17.2k12438
Your conjecture for GM was proved in 2011 in the short paper On a limit involving the product of prime numbers by József Sándor and Antoine Verroken.
Abstract. Let $p_k$ denote the $k$th prime number. The aim of this note is to prove that the limit of the sequence $(p_n / sqrt[n]p_1 cdots p_n)$ is $e$.
The authors obtain the result based on the prime number theorem, i.e.,
$$p approx n log n quad textrmas n to infty$$
as well as an inequality with Chebyshev's function $$theta(x) = sum_p le xlog p$$
where $p$ are primes less than $x$.
answered 12 mins ago
Théophile
17.2k12438
edited 1 min ago
answered 12 mins ago
Théophile
17.2k12438
answered 12 mins ago
Théophile
17.2k12438
answered 12 mins ago
Théophile
17.2k12438
17.2k12438
That contradicts my numerical observations. Could you please point out where I've gone wrong?
– TheSimpliFire
9 mins ago
@TheSimpliFire I think we simply can't conclude that the sequence is strictly decreasing based on the first $15$ values. In a paper from 2016 on the same topic, the author points out that the sequence converges slowly.
– Théophile
6 mins ago
add a comment |Â
That contradicts my numerical observations. Could you please point out where I've gone wrong?
– TheSimpliFire
9 mins ago
@TheSimpliFire I think we simply can't conclude that the sequence is strictly decreasing based on the first $15$ values. In a paper from 2016 on the same topic, the author points out that the sequence converges slowly.
– Théophile
6 mins ago
That contradicts my numerical observations. Could you please point out where I've gone wrong?
– TheSimpliFire
9 mins ago
That contradicts my numerical observations. Could you please point out where I've gone wrong?
– TheSimpliFire
9 mins ago
@TheSimpliFire I think we simply can't conclude that the sequence is strictly decreasing based on the first $15$ values. In a paper from 2016 on the same topic, the author points out that the sequence converges slowly.
– Théophile
6 mins ago
@TheSimpliFire I think we simply can't conclude that the sequence is strictly decreasing based on the first $15$ values. In a paper from 2016 on the same topic, the author points out that the sequence converges slowly.
– Théophile
6 mins ago
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2917887%2fdo-arithmetic-mean-and-geometric-mean-of-prime-numbers-converge%23new-answer', 'question_page');
);
Post as a guest
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
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
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
Can we take $p_n = ncdot log(n)$ in these limits? (using the prime numbers theorem)
– Yanko
22 mins ago