Values of Chebyshev’s ϑ-function for large numbers
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Is there website or article that has values of Chebyshev’s ϑ-function for very large numbers?
Chebyshev’s ϑ-function is
$$vartheta(x)=sum_{pleq x} ln p,$$
where $p$ is a prime number.
I have searched but I really couldn't find a resource that gives values of the function for numbers like $10^{19}$ or $10^{23}$.
prime-numbers
edited Nov 21 at 13:35
Klangen
1,36511231
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
add a comment |
up vote
0
down vote
favorite
Is there website or article that has values of Chebyshev’s ϑ-function for very large numbers?
Chebyshev’s ϑ-function is
$$vartheta(x)=sum_{pleq x} ln p,$$
where $p$ is a prime number.
I have searched but I really couldn't find a resource that gives values of the function for numbers like $10^{19}$ or $10^{23}$.
prime-numbers
edited Nov 21 at 13:35
Klangen
1,36511231
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
Only goes up to $10^{10}$ arxiv.org/pdf/1002.0442.pdf ... but you might find this interesting (sorry ?)
– Donald Splutterwit
Sep 4 '17 at 17:42
You could look for well-formatted prime tables and write a little program that computes the theta function from that, splitting the string at commata or something like that.
– Cloudscape
Sep 4 '17 at 17:54
Otherwise you'll find tables for $psi(x) = sum_{p^k le x} log p$. And you can construct it from the zeta zeros.
– reuns
Sep 4 '17 at 17:59
For smaller values, there are programs like Perl/ntheory that can produce arbitrary psi or theta values in reasonable times (e.g. 10 seconds for $10^{10}$). Scaling is close to linear so that makes $10^{19}$ not practical. Precision matters as well -- using native precision (e.g. 15 digits) is much faster than using extended precision (e.g. 40-100+ digits). Pari/GP could give the latter in a reasonable time for smaller values. Both via brute force. To get results for large values it seems you'd need to modify a fast prime count method (see prime sum methods using LMO / DR).
– DanaJ
Sep 4 '17 at 22:53
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Is there website or article that has values of Chebyshev’s ϑ-function for very large numbers?
Chebyshev’s ϑ-function is
$$vartheta(x)=sum_{pleq x} ln p,$$
where $p$ is a prime number.
I have searched but I really couldn't find a resource that gives values of the function for numbers like $10^{19}$ or $10^{23}$.
prime-numbers
edited Nov 21 at 13:35
Klangen
1,36511231
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
Is there website or article that has values of Chebyshev’s ϑ-function for very large numbers?
Chebyshev’s ϑ-function is
$$vartheta(x)=sum_{pleq x} ln p,$$
where $p$ is a prime number.
I have searched but I really couldn't find a resource that gives values of the function for numbers like $10^{19}$ or $10^{23}$.
prime-numbers
prime-numbers
edited Nov 21 at 13:35
Klangen
1,36511231
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
edited Nov 21 at 13:35
Klangen
1,36511231
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
edited Nov 21 at 13:35
Klangen
1,36511231
edited Nov 21 at 13:35
Klangen
1,36511231
edited Nov 21 at 13:35
Klangen
1,36511231
1,36511231
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
asked Sep 4 '17 at 17:36
Bright Chancellor
351311
351311
Only goes up to $10^{10}$ arxiv.org/pdf/1002.0442.pdf ... but you might find this interesting (sorry ?)
– Donald Splutterwit
Sep 4 '17 at 17:42
You could look for well-formatted prime tables and write a little program that computes the theta function from that, splitting the string at commata or something like that.
– Cloudscape
Sep 4 '17 at 17:54
Otherwise you'll find tables for $psi(x) = sum_{p^k le x} log p$. And you can construct it from the zeta zeros.
– reuns
Sep 4 '17 at 17:59
For smaller values, there are programs like Perl/ntheory that can produce arbitrary psi or theta values in reasonable times (e.g. 10 seconds for $10^{10}$). Scaling is close to linear so that makes $10^{19}$ not practical. Precision matters as well -- using native precision (e.g. 15 digits) is much faster than using extended precision (e.g. 40-100+ digits). Pari/GP could give the latter in a reasonable time for smaller values. Both via brute force. To get results for large values it seems you'd need to modify a fast prime count method (see prime sum methods using LMO / DR).
– DanaJ
Sep 4 '17 at 22:53
add a comment |
Only goes up to $10^{10}$ arxiv.org/pdf/1002.0442.pdf ... but you might find this interesting (sorry ?)
– Donald Splutterwit
Sep 4 '17 at 17:42
You could look for well-formatted prime tables and write a little program that computes the theta function from that, splitting the string at commata or something like that.
– Cloudscape
Sep 4 '17 at 17:54
Otherwise you'll find tables for $psi(x) = sum_{p^k le x} log p$. And you can construct it from the zeta zeros.
– reuns
Sep 4 '17 at 17:59
For smaller values, there are programs like Perl/ntheory that can produce arbitrary psi or theta values in reasonable times (e.g. 10 seconds for $10^{10}$). Scaling is close to linear so that makes $10^{19}$ not practical. Precision matters as well -- using native precision (e.g. 15 digits) is much faster than using extended precision (e.g. 40-100+ digits). Pari/GP could give the latter in a reasonable time for smaller values. Both via brute force. To get results for large values it seems you'd need to modify a fast prime count method (see prime sum methods using LMO / DR).
– DanaJ
Sep 4 '17 at 22:53
Only goes up to $10^{10}$ arxiv.org/pdf/1002.0442.pdf ... but you might find this interesting (sorry ?)
– Donald Splutterwit
Sep 4 '17 at 17:42
Only goes up to $10^{10}$ arxiv.org/pdf/1002.0442.pdf ... but you might find this interesting (sorry ?)
– Donald Splutterwit
Sep 4 '17 at 17:42
You could look for well-formatted prime tables and write a little program that computes the theta function from that, splitting the string at commata or something like that.
– Cloudscape
Sep 4 '17 at 17:54
You could look for well-formatted prime tables and write a little program that computes the theta function from that, splitting the string at commata or something like that.
– Cloudscape
Sep 4 '17 at 17:54
Otherwise you'll find tables for $psi(x) = sum_{p^k le x} log p$. And you can construct it from the zeta zeros.
– reuns
Sep 4 '17 at 17:59
Otherwise you'll find tables for $psi(x) = sum_{p^k le x} log p$. And you can construct it from the zeta zeros.
– reuns
Sep 4 '17 at 17:59
For smaller values, there are programs like Perl/ntheory that can produce arbitrary psi or theta values in reasonable times (e.g. 10 seconds for $10^{10}$). Scaling is close to linear so that makes $10^{19}$ not practical. Precision matters as well -- using native precision (e.g. 15 digits) is much faster than using extended precision (e.g. 40-100+ digits). Pari/GP could give the latter in a reasonable time for smaller values. Both via brute force. To get results for large values it seems you'd need to modify a fast prime count method (see prime sum methods using LMO / DR).
– DanaJ
Sep 4 '17 at 22:53
For smaller values, there are programs like Perl/ntheory that can produce arbitrary psi or theta values in reasonable times (e.g. 10 seconds for $10^{10}$). Scaling is close to linear so that makes $10^{19}$ not practical. Precision matters as well -- using native precision (e.g. 15 digits) is much faster than using extended precision (e.g. 40-100+ digits). Pari/GP could give the latter in a reasonable time for smaller values. Both via brute force. To get results for large values it seems you'd need to modify a fast prime count method (see prime sum methods using LMO / DR).
– DanaJ
Sep 4 '17 at 22:53
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Only goes up to $10^{10}$ arxiv.org/pdf/1002.0442.pdf ... but you might find this interesting (sorry ?)
– Donald Splutterwit
Sep 4 '17 at 17:42
You could look for well-formatted prime tables and write a little program that computes the theta function from that, splitting the string at commata or something like that.
– Cloudscape
Sep 4 '17 at 17:54
Otherwise you'll find tables for $psi(x) = sum_{p^k le x} log p$. And you can construct it from the zeta zeros.
– reuns
Sep 4 '17 at 17:59
For smaller values, there are programs like Perl/ntheory that can produce arbitrary psi or theta values in reasonable times (e.g. 10 seconds for $10^{10}$). Scaling is close to linear so that makes $10^{19}$ not practical. Precision matters as well -- using native precision (e.g. 15 digits) is much faster than using extended precision (e.g. 40-100+ digits). Pari/GP could give the latter in a reasonable time for smaller values. Both via brute force. To get results for large values it seems you'd need to modify a fast prime count method (see prime sum methods using LMO / DR).
– DanaJ
Sep 4 '17 at 22:53