Mixing
Strengthening Bertrand's postulate using the prime number theorem
$begingroup$
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
28.8k95296
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
$endgroup$
add a comment |
$begingroup$
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
28.8k95296
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
$endgroup$
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
$begingroup$
I think it better to close-as-duplicate, see How far to nearest/next prime? where the answers gives some formal references.
$endgroup$
– hardmath
Jan 2 at 18:22
$begingroup$
Possible duplicate of How far to nearest/next prime?
$endgroup$
– hardmath
Jan 2 at 18:23
add a comment |
$begingroup$
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
28.8k95296
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
$endgroup$
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
reference-request analytic-number-theory
edited Jan 2 at 18:23
hardmath
28.8k95296
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
edited Jan 2 at 18:23
hardmath
28.8k95296
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
edited Jan 2 at 18:23
hardmath
28.8k95296
edited Jan 2 at 18:23
hardmath
28.8k95296
edited Jan 2 at 18:23
hardmath
28.8k95296
28.8k95296
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
asked Jul 15 '18 at 23:54
Ethan BolkerEthan Bolker
42.1k548111
42.1k548111
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
$begingroup$
I think it better to close-as-duplicate, see How far to nearest/next prime? where the answers gives some formal references.
$endgroup$
– hardmath
Jan 2 at 18:22
$begingroup$
Possible duplicate of How far to nearest/next prime?
$endgroup$
– hardmath
Jan 2 at 18:23
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
$begingroup$
I think it better to close-as-duplicate, see How far to nearest/next prime? where the answers gives some formal references.
$endgroup$
– hardmath
Jan 2 at 18:22
$begingroup$
Possible duplicate of How far to nearest/next prime?
$endgroup$
– hardmath
Jan 2 at 18:23
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
$begingroup$
I think it better to close-as-duplicate, see How far to nearest/next prime? where the answers gives some formal references.
$endgroup$
– hardmath
Jan 2 at 18:22
$begingroup$
I think it better to close-as-duplicate, see How far to nearest/next prime? where the answers gives some formal references.
$endgroup$
– hardmath
Jan 2 at 18:22
$begingroup$
Possible duplicate of How far to nearest/next prime?
$endgroup$
– hardmath
Jan 2 at 18:23
$begingroup$
Possible duplicate of How far to nearest/next prime?
$endgroup$
– hardmath
Jan 2 at 18:23
add a comment |
2 Answers
2
active
oldest
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$begingroup$
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
68.7k22685
$endgroup$
$begingroup$
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
102k5101199
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
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
68.7k22685
$endgroup$
$begingroup$
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
68.7k22685
$endgroup$
$begingroup$
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
68.7k22685
$endgroup$
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
68.7k22685
edited Jul 16 '18 at 2:28
answered Jul 16 '18 at 0:12
fleabloodfleablood
68.7k22685
answered Jul 16 '18 at 0:12
fleabloodfleablood
68.7k22685
answered Jul 16 '18 at 0:12
fleabloodfleablood
68.7k22685
68.7k22685
$begingroup$
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
102k5101199
$endgroup$
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
102k5101199
$endgroup$
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
102k5101199
$endgroup$
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
102k5101199
edited Jul 16 '18 at 0:38
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
102k5101199
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
102k5101199
answered Jul 16 '18 at 0:22
Will JagyWill Jagy
102k5101199
102k5101199
add a comment |
add a comment |
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$begingroup$
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
$begingroup$
I think it better to close-as-duplicate, see How far to nearest/next prime? where the answers gives some formal references.
$endgroup$
– hardmath
Jan 2 at 18:22
$begingroup$
Possible duplicate of How far to nearest/next prime?
$endgroup$
– hardmath
Jan 2 at 18:23