Estimating the Twin prime constant
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On this website:
http://numbers.computation.free.fr/Constants/Primes/twin.html
it says:
"This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be estimated using properties of the Riemann Zeta function to thousand of digits (Sebah computed it to more than 5000 digits)."
Can someone explain how you can use the Riemann zeta function to estimate the digits of the twin prime constant?
number-theory prime-numbers prime-twins
edited Dec 17 '18 at 10:37
Klangen
1,73711334
asked May 14 '14 at 17:37
user146828user146828
284
$endgroup$
add a comment |
$begingroup$
On this website:
http://numbers.computation.free.fr/Constants/Primes/twin.html
it says:
"This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be estimated using properties of the Riemann Zeta function to thousand of digits (Sebah computed it to more than 5000 digits)."
Can someone explain how you can use the Riemann zeta function to estimate the digits of the twin prime constant?
number-theory prime-numbers prime-twins
edited Dec 17 '18 at 10:37
Klangen
1,73711334
asked May 14 '14 at 17:37
user146828user146828
284
$endgroup$
1
$begingroup$
Did you try the reference? T. Nicely, Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant, Virginia J. Sci., (1996), vol. 46, p. 195-204
$endgroup$
– Matthew Conroy
May 14 '14 at 17:39
1
$begingroup$
Perhaps even better: trnicely.net/twins/twins2.html
$endgroup$
– Matthew Conroy
May 14 '14 at 17:41
add a comment |
$begingroup$
On this website:
http://numbers.computation.free.fr/Constants/Primes/twin.html
it says:
"This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be estimated using properties of the Riemann Zeta function to thousand of digits (Sebah computed it to more than 5000 digits)."
Can someone explain how you can use the Riemann zeta function to estimate the digits of the twin prime constant?
number-theory prime-numbers prime-twins
edited Dec 17 '18 at 10:37
Klangen
1,73711334
asked May 14 '14 at 17:37
user146828user146828
284
$endgroup$
On this website:
http://numbers.computation.free.fr/Constants/Primes/twin.html
it says:
"This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be estimated using properties of the Riemann Zeta function to thousand of digits (Sebah computed it to more than 5000 digits)."
Can someone explain how you can use the Riemann zeta function to estimate the digits of the twin prime constant?
number-theory prime-numbers prime-twins
number-theory prime-numbers prime-twins
edited Dec 17 '18 at 10:37
Klangen
1,73711334
asked May 14 '14 at 17:37
user146828user146828
284
edited Dec 17 '18 at 10:37
Klangen
1,73711334
asked May 14 '14 at 17:37
user146828user146828
284
edited Dec 17 '18 at 10:37
Klangen
1,73711334
edited Dec 17 '18 at 10:37
Klangen
1,73711334
edited Dec 17 '18 at 10:37
Klangen
1,73711334
1,73711334
asked May 14 '14 at 17:37
user146828user146828
284
asked May 14 '14 at 17:37
user146828user146828
284
asked May 14 '14 at 17:37
user146828user146828
284
284
1
$begingroup$
Did you try the reference? T. Nicely, Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant, Virginia J. Sci., (1996), vol. 46, p. 195-204
$endgroup$
– Matthew Conroy
May 14 '14 at 17:39
1
$begingroup$
Perhaps even better: trnicely.net/twins/twins2.html
$endgroup$
– Matthew Conroy
May 14 '14 at 17:41
add a comment |
1
$begingroup$
Did you try the reference? T. Nicely, Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant, Virginia J. Sci., (1996), vol. 46, p. 195-204
$endgroup$
– Matthew Conroy
May 14 '14 at 17:39
1
$begingroup$
Perhaps even better: trnicely.net/twins/twins2.html
$endgroup$
– Matthew Conroy
May 14 '14 at 17:41
1
1
$begingroup$
Did you try the reference? T. Nicely, Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant, Virginia J. Sci., (1996), vol. 46, p. 195-204
$endgroup$
– Matthew Conroy
May 14 '14 at 17:39
$begingroup$
Did you try the reference? T. Nicely, Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant, Virginia J. Sci., (1996), vol. 46, p. 195-204
$endgroup$
– Matthew Conroy
May 14 '14 at 17:39
1
1
$begingroup$
Perhaps even better: trnicely.net/twins/twins2.html
$endgroup$
– Matthew Conroy
May 14 '14 at 17:41
$begingroup$
Perhaps even better: trnicely.net/twins/twins2.html
$endgroup$
– Matthew Conroy
May 14 '14 at 17:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
There exists a formula that gives a series with accelerated convergence for the twin prime constant, due to Flajolet and Vardi (1996). If we denote the twin prime constant by $Pi_2$, we have:
$$
Pi_2 = prod_{n=2}^{infty}left[zeta(n)(1-2^{-n})right]^{-I_n},
$$
where
$$
I_n=frac{1}{n}sum_{d|n}mu(d)2^{n/d}.
$$
The values of $I_n$ can be found in the OEIS as sequence A001037:
$$
I_n = 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ldots
$$
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
$endgroup$
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
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active
oldest
votes
$begingroup$
There exists a formula that gives a series with accelerated convergence for the twin prime constant, due to Flajolet and Vardi (1996). If we denote the twin prime constant by $Pi_2$, we have:
$$
Pi_2 = prod_{n=2}^{infty}left[zeta(n)(1-2^{-n})right]^{-I_n},
$$
where
$$
I_n=frac{1}{n}sum_{d|n}mu(d)2^{n/d}.
$$
The values of $I_n$ can be found in the OEIS as sequence A001037:
$$
I_n = 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ldots
$$
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
$endgroup$
add a comment |
$begingroup$
There exists a formula that gives a series with accelerated convergence for the twin prime constant, due to Flajolet and Vardi (1996). If we denote the twin prime constant by $Pi_2$, we have:
$$
Pi_2 = prod_{n=2}^{infty}left[zeta(n)(1-2^{-n})right]^{-I_n},
$$
where
$$
I_n=frac{1}{n}sum_{d|n}mu(d)2^{n/d}.
$$
The values of $I_n$ can be found in the OEIS as sequence A001037:
$$
I_n = 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ldots
$$
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
$endgroup$
add a comment |
$begingroup$
There exists a formula that gives a series with accelerated convergence for the twin prime constant, due to Flajolet and Vardi (1996). If we denote the twin prime constant by $Pi_2$, we have:
$$
Pi_2 = prod_{n=2}^{infty}left[zeta(n)(1-2^{-n})right]^{-I_n},
$$
where
$$
I_n=frac{1}{n}sum_{d|n}mu(d)2^{n/d}.
$$
The values of $I_n$ can be found in the OEIS as sequence A001037:
$$
I_n = 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ldots
$$
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
$endgroup$
There exists a formula that gives a series with accelerated convergence for the twin prime constant, due to Flajolet and Vardi (1996). If we denote the twin prime constant by $Pi_2$, we have:
$$
Pi_2 = prod_{n=2}^{infty}left[zeta(n)(1-2^{-n})right]^{-I_n},
$$
where
$$
I_n=frac{1}{n}sum_{d|n}mu(d)2^{n/d}.
$$
The values of $I_n$ can be found in the OEIS as sequence A001037:
$$
I_n = 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ldots
$$
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
answered Dec 17 '18 at 10:02
KlangenKlangen
1,73711334
1,73711334
add a comment |
add a comment |
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$begingroup$
Did you try the reference? T. Nicely, Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant, Virginia J. Sci., (1996), vol. 46, p. 195-204
$endgroup$
– Matthew Conroy
May 14 '14 at 17:39
1
$begingroup$
Perhaps even better: trnicely.net/twins/twins2.html
$endgroup$
– Matthew Conroy
May 14 '14 at 17:41