How accurately must I compute the twin prime constant to get the twin prime density?
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Let $pi _{2}(x)$ denote the number of primes $pleq x$ such that $p+2$ is also prime. Hardy and Littlewood conjectured that
$$
{displaystyle pi _{2}(x)sim 2C_{2}{frac {x}{(ln x)^{2}}}sim 2C_{2}int _{2}^{x}{dt over (ln t)^{2}}},
$$
where
$$
{displaystyle C_{2}=prod _{textstyle {p;{rm {prime}} atop pgeq 3}}left(1-{frac {1}{(p-1)^{2}}}right)approx 0.660161815846869573927812110014dots }.
$$
I want to numerically study this density, so I implemented some C++ code that computes the above value for $pi_2(x)$. However, before launching any computations I want to make sure that I set everything up correctly. In particular, I am unsure about what precision I should use for $C_2$, i.e., the twin-prime constant, in function of my input $x$.
In other words, how many primes should I include within the product for $C_2$ so that I get the correct expected conjectured density for $pi_2(x)$? For example, if I want to calculate the conjectured density for $pi_2(10^5)$, should I include all primes $leq 10^5$ in the product for $C_2$?
asymptotics computational-mathematics prime-twins
asked Dec 1 at 12:41
Klangen
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add a comment |
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Let $pi _{2}(x)$ denote the number of primes $pleq x$ such that $p+2$ is also prime. Hardy and Littlewood conjectured that
$$
{displaystyle pi _{2}(x)sim 2C_{2}{frac {x}{(ln x)^{2}}}sim 2C_{2}int _{2}^{x}{dt over (ln t)^{2}}},
$$
where
$$
{displaystyle C_{2}=prod _{textstyle {p;{rm {prime}} atop pgeq 3}}left(1-{frac {1}{(p-1)^{2}}}right)approx 0.660161815846869573927812110014dots }.
$$
I want to numerically study this density, so I implemented some C++ code that computes the above value for $pi_2(x)$. However, before launching any computations I want to make sure that I set everything up correctly. In particular, I am unsure about what precision I should use for $C_2$, i.e., the twin-prime constant, in function of my input $x$.
In other words, how many primes should I include within the product for $C_2$ so that I get the correct expected conjectured density for $pi_2(x)$? For example, if I want to calculate the conjectured density for $pi_2(10^5)$, should I include all primes $leq 10^5$ in the product for $C_2$?
asymptotics computational-mathematics prime-twins
asked Dec 1 at 12:41
Klangen
1,25811129
This question has an open bounty worth +50
reputation from Klangen ending in 6 days.
This question has not received enough attention.
The number of digits of $C_2$ given in the problem will be more than accurate enough for any $x$ you can reach numerically. Sub-leading terms in the asymptotic form are going to contribute much earlier than the 30th significant figure.
– mjqxxxx
yesterday
@mjqxxxx Thank you for your comment. Can you give a more detailed answer, in terms of $x$? For instance, how many primes should I consider in the product for $pi_2(10^x)$?
– Klangen
yesterday
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $pi _{2}(x)$ denote the number of primes $pleq x$ such that $p+2$ is also prime. Hardy and Littlewood conjectured that
$$
{displaystyle pi _{2}(x)sim 2C_{2}{frac {x}{(ln x)^{2}}}sim 2C_{2}int _{2}^{x}{dt over (ln t)^{2}}},
$$
where
$$
{displaystyle C_{2}=prod _{textstyle {p;{rm {prime}} atop pgeq 3}}left(1-{frac {1}{(p-1)^{2}}}right)approx 0.660161815846869573927812110014dots }.
$$
I want to numerically study this density, so I implemented some C++ code that computes the above value for $pi_2(x)$. However, before launching any computations I want to make sure that I set everything up correctly. In particular, I am unsure about what precision I should use for $C_2$, i.e., the twin-prime constant, in function of my input $x$.
In other words, how many primes should I include within the product for $C_2$ so that I get the correct expected conjectured density for $pi_2(x)$? For example, if I want to calculate the conjectured density for $pi_2(10^5)$, should I include all primes $leq 10^5$ in the product for $C_2$?
asymptotics computational-mathematics prime-twins
asked Dec 1 at 12:41
Klangen
1,25811129
Let $pi _{2}(x)$ denote the number of primes $pleq x$ such that $p+2$ is also prime. Hardy and Littlewood conjectured that
$$
{displaystyle pi _{2}(x)sim 2C_{2}{frac {x}{(ln x)^{2}}}sim 2C_{2}int _{2}^{x}{dt over (ln t)^{2}}},
$$
where
$$
{displaystyle C_{2}=prod _{textstyle {p;{rm {prime}} atop pgeq 3}}left(1-{frac {1}{(p-1)^{2}}}right)approx 0.660161815846869573927812110014dots }.
$$
I want to numerically study this density, so I implemented some C++ code that computes the above value for $pi_2(x)$. However, before launching any computations I want to make sure that I set everything up correctly. In particular, I am unsure about what precision I should use for $C_2$, i.e., the twin-prime constant, in function of my input $x$.
In other words, how many primes should I include within the product for $C_2$ so that I get the correct expected conjectured density for $pi_2(x)$? For example, if I want to calculate the conjectured density for $pi_2(10^5)$, should I include all primes $leq 10^5$ in the product for $C_2$?
asymptotics computational-mathematics prime-twins
asymptotics computational-mathematics prime-twins
asked Dec 1 at 12:41
Klangen
1,25811129
asked Dec 1 at 12:41
Klangen
1,25811129
asked Dec 1 at 12:41
Klangen
1,25811129
asked Dec 1 at 12:41
Klangen
1,25811129
asked Dec 1 at 12:41
Klangen
1,25811129
1,25811129
This question has an open bounty worth +50
reputation from Klangen ending in 6 days.
This question has not received enough attention.
This question has an open bounty worth +50
reputation from Klangen ending in 6 days.
This question has not received enough attention.
The number of digits of $C_2$ given in the problem will be more than accurate enough for any $x$ you can reach numerically. Sub-leading terms in the asymptotic form are going to contribute much earlier than the 30th significant figure.
– mjqxxxx
yesterday
@mjqxxxx Thank you for your comment. Can you give a more detailed answer, in terms of $x$? For instance, how many primes should I consider in the product for $pi_2(10^x)$?
– Klangen
yesterday
add a comment |
The number of digits of $C_2$ given in the problem will be more than accurate enough for any $x$ you can reach numerically. Sub-leading terms in the asymptotic form are going to contribute much earlier than the 30th significant figure.
– mjqxxxx
yesterday
@mjqxxxx Thank you for your comment. Can you give a more detailed answer, in terms of $x$? For instance, how many primes should I consider in the product for $pi_2(10^x)$?
– Klangen
yesterday
The number of digits of $C_2$ given in the problem will be more than accurate enough for any $x$ you can reach numerically. Sub-leading terms in the asymptotic form are going to contribute much earlier than the 30th significant figure.
– mjqxxxx
yesterday
The number of digits of $C_2$ given in the problem will be more than accurate enough for any $x$ you can reach numerically. Sub-leading terms in the asymptotic form are going to contribute much earlier than the 30th significant figure.
– mjqxxxx
yesterday
@mjqxxxx Thank you for your comment. Can you give a more detailed answer, in terms of $x$? For instance, how many primes should I consider in the product for $pi_2(10^x)$?
– Klangen
yesterday
@mjqxxxx Thank you for your comment. Can you give a more detailed answer, in terms of $x$? For instance, how many primes should I consider in the product for $pi_2(10^x)$?
– Klangen
yesterday
add a comment |
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The number of digits of $C_2$ given in the problem will be more than accurate enough for any $x$ you can reach numerically. Sub-leading terms in the asymptotic form are going to contribute much earlier than the 30th significant figure.
– mjqxxxx
yesterday
@mjqxxxx Thank you for your comment. Can you give a more detailed answer, in terms of $x$? For instance, how many primes should I consider in the product for $pi_2(10^x)$?
– Klangen
yesterday