Strengthening Bertrand's postulate using the prime number theorem [duplicate]
$begingroup$
This question already has an answer here:
How far to nearest/next prime?
1 answer
In its page on Bertrand's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
reference-request analytic-number-theory
edited Jan 2 at 18:23
hardmath
29.1k953101
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
$endgroup$
marked as duplicate by hardmath, user91500, Lord_Farin, Cesareo, Jyrki Lahtonen Jan 17 at 15:15
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
How far to nearest/next prime?
1 answer
In its page on Bertrand's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
reference-request analytic-number-theory
edited Jan 2 at 18:23
hardmath
29.1k953101
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
$endgroup$
marked as duplicate by hardmath, user91500, Lord_Farin, Cesareo, Jyrki Lahtonen Jan 17 at 15:15
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
$begingroup$
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
$endgroup$
– fleablood
Jul 16 '18 at 0:00
$begingroup$
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
$endgroup$
– Ethan Bolker
Jul 16 '18 at 0:04
add a comment |
$begingroup$
This question already has an answer here:
How far to nearest/next prime?
1 answer
In its page on Bertrand's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
reference-request analytic-number-theory
edited Jan 2 at 18:23
hardmath
29.1k953101
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
$endgroup$
This question already has an answer here:
How far to nearest/next prime?
1 answer
In its page on Bertrand's postulate Wikipedia says
It follows from the prime number theorem that for any real
$varepsilon >0$ there is an $n_0 > 0$ such that for all $n > n_0$
there is a prime $p $ such that $n < p < ( 1 + ε )n$ .
I need to quote this result for $varepsilon = 1/2$. I'd like a reference more formal than Wikipedia.
This question already has an answer here:
How far to nearest/next prime?
1 answer
reference-request analytic-number-theory
reference-request analytic-number-theory
edited Jan 2 at 18:23
hardmath
29.1k953101
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
edited Jan 2 at 18:23
hardmath
29.1k953101
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
edited Jan 2 at 18:23
hardmath
29.1k953101
edited Jan 2 at 18:23
hardmath
29.1k953101
edited Jan 2 at 18:23
hardmath
29.1k953101
29.1k953101
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
44.7k553120
44.7k553120
marked as duplicate by hardmath, user91500, Lord_Farin, Cesareo, Jyrki Lahtonen Jan 17 at 15:15
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by hardmath, user91500, Lord_Farin, Cesareo, Jyrki Lahtonen Jan 17 at 15:15
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
$begingroup$
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
$endgroup$
– fleablood
Jul 16 '18 at 0:00
$begingroup$
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
$endgroup$
– Ethan Bolker
Jul 16 '18 at 0:04
add a comment |
1
$begingroup$
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
$endgroup$
– fleablood
Jul 16 '18 at 0:00
$begingroup$
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
$endgroup$
– Ethan Bolker
Jul 16 '18 at 0:04
1
1
$begingroup$
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
$endgroup$
– fleablood
Jul 16 '18 at 0:00
$begingroup$
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
$endgroup$
– fleablood
Jul 16 '18 at 0:00
$begingroup$
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
$endgroup$
– Ethan Bolker
Jul 16 '18 at 0:04
$begingroup$
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
$endgroup$
– Ethan Bolker
Jul 16 '18 at 0:04
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ {displaystyle varepsilon >0}$ there is a ${displaystyle n_{0}>0} $ such that for all ${displaystyle n>n_{0}} $ there is a prime $ {displaystyle p}$ such ${displaystyle n<p<(1+varepsilon )n}$. It can be shown, for instance,
that
${displaystyle lim _{nto infty }{frac {pi ((1+varepsilon )n)-pi
> (n)}{n/log n}}=varepsilon ,} $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
$endgroup$
$begingroup$
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
$endgroup$
– Steve Kass
Jul 16 '18 at 0:34
$begingroup$
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
$endgroup$
– fleablood
Jul 16 '18 at 2:19
add a comment |
$begingroup$
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac{1}{25 log^2 x} right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^{18},$ we get gap $g < log^2 p.$
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
$endgroup$
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ {displaystyle varepsilon >0}$ there is a ${displaystyle n_{0}>0} $ such that for all ${displaystyle n>n_{0}} $ there is a prime $ {displaystyle p}$ such ${displaystyle n<p<(1+varepsilon )n}$. It can be shown, for instance,
that
${displaystyle lim _{nto infty }{frac {pi ((1+varepsilon )n)-pi
> (n)}{n/log n}}=varepsilon ,} $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
$endgroup$
$begingroup$
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
$endgroup$
– Steve Kass
Jul 16 '18 at 0:34
$begingroup$
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
$endgroup$
– fleablood
Jul 16 '18 at 2:19
add a comment |
$begingroup$
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ {displaystyle varepsilon >0}$ there is a ${displaystyle n_{0}>0} $ such that for all ${displaystyle n>n_{0}} $ there is a prime $ {displaystyle p}$ such ${displaystyle n<p<(1+varepsilon )n}$. It can be shown, for instance,
that
${displaystyle lim _{nto infty }{frac {pi ((1+varepsilon )n)-pi
> (n)}{n/log n}}=varepsilon ,} $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
$endgroup$
$begingroup$
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
$endgroup$
– Steve Kass
Jul 16 '18 at 0:34
$begingroup$
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
$endgroup$
– fleablood
Jul 16 '18 at 2:19
add a comment |
$begingroup$
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ {displaystyle varepsilon >0}$ there is a ${displaystyle n_{0}>0} $ such that for all ${displaystyle n>n_{0}} $ there is a prime $ {displaystyle p}$ such ${displaystyle n<p<(1+varepsilon )n}$. It can be shown, for instance,
that
${displaystyle lim _{nto infty }{frac {pi ((1+varepsilon )n)-pi
> (n)}{n/log n}}=varepsilon ,} $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
$endgroup$
The ariticle on wikipededia cites that with a foot note (as of the time I write this) #8 refering to: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
from Wikipedia
It follows from the prime number theorem that for any real $ {displaystyle varepsilon >0}$ there is a ${displaystyle n_{0}>0} $ such that for all ${displaystyle n>n_{0}} $ there is a prime $ {displaystyle p}$ such ${displaystyle n<p<(1+varepsilon )n}$. It can be shown, for instance,
that
${displaystyle lim _{nto infty }{frac {pi ((1+varepsilon )n)-pi
> (n)}{n/log n}}=varepsilon ,} $[8]
[8]G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
edited Jul 16 '18 at 2:28
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
answered Jul 16 '18 at 0:12
fleabloodfleablood
71.9k22687
71.9k22687
$begingroup$
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
$endgroup$
– Steve Kass
Jul 16 '18 at 0:34
$begingroup$
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
$endgroup$
– fleablood
Jul 16 '18 at 2:19
add a comment |
$begingroup$
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
$endgroup$
– Steve Kass
Jul 16 '18 at 0:34
$begingroup$
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
$endgroup$
– fleablood
Jul 16 '18 at 2:19
$begingroup$
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
$endgroup$
– Steve Kass
Jul 16 '18 at 0:34
$begingroup$
This is pretty unreadable. I doubt that Hardy and Wright says “for any real $epsilon > 0 > epsilon > 0epsilon > 0$, for example (and it doesn’t get better). Can you edit the quote so it doesn’t look like a copy/paste from some non-LaTeX source? Even an image of the relevant Hardy and Wright page would be better than this.
$endgroup$
– Steve Kass
Jul 16 '18 at 0:34
$begingroup$
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
$endgroup$
– fleablood
Jul 16 '18 at 2:19
$begingroup$
It is a copy and paste from a non-LaTeX source! I thought the was clear from context.
$endgroup$
– fleablood
Jul 16 '18 at 2:19
add a comment |
$begingroup$
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac{1}{25 log^2 x} right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^{18},$ we get gap $g < log^2 p.$
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
$endgroup$
add a comment |
$begingroup$
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac{1}{25 log^2 x} right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^{18},$ we get gap $g < log^2 p.$
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
$endgroup$
add a comment |
$begingroup$
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac{1}{25 log^2 x} right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^{18},$ we get gap $g < log^2 p.$
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
$endgroup$
tersely worded, but Dusart gives, for $x geq 396738,$ a prime between $x$ and
$$ x left( 1 + frac{1}{25 log^2 x} right) $$
I think it was on the arxiv, let me find it.
Yes, this is Proposition 6.8.
For your fixed target $3x/2,$ you may fill in the smaller numbers with the table of maximal prime gaps at https://en.wikipedia.org/wiki/Prime_gap#Numerical_results . I went through the thing once by machine, for $p geq 11$ and $p < 4 cdot 10^{18},$ we get gap $g < log^2 p.$
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
edited Jul 16 '18 at 0:38
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
104k5102201
104k5102201
add a comment |
add a comment |
1
$begingroup$
Why don't you cite the reference that very source that Wikipedia used: G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, p. 494.
$endgroup$
– fleablood
Jul 16 '18 at 0:00
$begingroup$
@fleablood Of course. I missed that reference. Post as an answer, or I can delete the question as too narrow.
$endgroup$
– Ethan Bolker
Jul 16 '18 at 0:04