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Bounds for $n$-th prime

How do I find upper bound for nth prime number?Size of the $k$-th prime numberLower and upper bounds for the $n$th prime numberProve if $ngeq 9$ than $P_n> 2n+3$, p is a primeDoes this limit involving the Dirichlet eta function and the Riemann zeta function make sense?Show that $pi(p_x^2) geq 2p_x -2$Question on convergence of sum(prime(n)/prime(n+1)/n^2,n=1…infinity)Smallest prime in arithmetic progressions: upper bounds?Simplest nontrivial example repositoryList of Bounds of $n$-th compositeProof of $p_n<n^2$ by Elementary MeansInequality $p_n<nleft(log n + loglog nright)$Is there always a prime between a prime and prime plus the index of that prime?Find an $n$ for which $(P_1,dots,P_n)+1$ is not primeWhat is the Largest Possible Prime Number? sic The largest possible prime set?Prime number theorem $n^rm th$ prime boundsUpper and Lower bounds on the nth prime number

13

6

$begingroup$

In this Wikipedia page I have found that the bounds for $n$-th prime is given by, $$n(ln n+ln ln n)>p_n>n(ln n+ln ln n-1)$$ for all $nge6$. Are there even stronger bounds for the $n$-th prime?

If possible (of course if the answer is affirmative) in the answer (or comment) please give the link of the paper in which it first appears.

prime-numbers big-list

share|cite|improve this question

edited Dec 31 '15 at 13:23

Alex M.

28.2k103159

asked May 7 '15 at 3:25

user 170039user 170039

10.5k42467

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  projecteuclid.org/euclid.ijm/1255631807
  $endgroup$
  – Will Jagy
  May 7 '15 at 3:36
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  See the work of Pierre Dusart: Dusart (1999): The kth prime is greater than k(log k + log log k - 1) for k>=2 (ams.org/journals/mcom/1999-68-225/S0025-5718-99-01037-6), Dusart (1998): Sharper bounds for $psi,theta,pi,p_k$: (unilim.fr/laco/rapports/1998/R1998_06.pdf)
  $endgroup$
  – gammatester
  May 7 '15 at 8:31
  

  
  
  
  

add a comment | 

13

6

$begingroup$

In this Wikipedia page I have found that the bounds for $n$-th prime is given by, $$n(ln n+ln ln n)>p_n>n(ln n+ln ln n-1)$$ for all $nge6$. Are there even stronger bounds for the $n$-th prime?

If possible (of course if the answer is affirmative) in the answer (or comment) please give the link of the paper in which it first appears.

prime-numbers big-list

share|cite|improve this question

edited Dec 31 '15 at 13:23

Alex M.

28.2k103159

asked May 7 '15 at 3:25

user 170039user 170039

10.5k42467

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  projecteuclid.org/euclid.ijm/1255631807
  $endgroup$
  – Will Jagy
  May 7 '15 at 3:36
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  See the work of Pierre Dusart: Dusart (1999): The kth prime is greater than k(log k + log log k - 1) for k>=2 (ams.org/journals/mcom/1999-68-225/S0025-5718-99-01037-6), Dusart (1998): Sharper bounds for $psi,theta,pi,p_k$: (unilim.fr/laco/rapports/1998/R1998_06.pdf)
  $endgroup$
  – gammatester
  May 7 '15 at 8:31
  

  
  
  
  

add a comment | 

$begingroup$

In this Wikipedia page I have found that the bounds for $n$-th prime is given by, $$n(ln n+ln ln n)>p_n>n(ln n+ln ln n-1)$$ for all $nge6$. Are there even stronger bounds for the $n$-th prime?

If possible (of course if the answer is affirmative) in the answer (or comment) please give the link of the paper in which it first appears.

prime-numbers big-list

share|cite|improve this question

edited Dec 31 '15 at 13:23

Alex M.

28.2k103159

asked May 7 '15 at 3:25

user 170039user 170039

10.5k42467

$endgroup$

share|cite|improve this question

edited Dec 31 '15 at 13:23

Alex M.

28.2k103159

asked May 7 '15 at 3:25

user 170039user 170039

10.5k42467

share|cite|improve this question

edited Dec 31 '15 at 13:23

Alex M.

28.2k103159

asked May 7 '15 at 3:25

user 170039user 170039

10.5k42467

edited Dec 31 '15 at 13:23

Alex M.

28.2k103159

edited Dec 31 '15 at 13:23

Alex M.

28.2k103159

asked May 7 '15 at 3:25

user 170039user 170039

10.5k42467

asked May 7 '15 at 3:25

user 170039user 170039

10.5k42467

1 Answer
1

active

oldest

votes

12

$begingroup$

See the papers of:

• Dusart 2010


• Axler 2013


• Axler 2014


• Büthe 2014


• Kotnick 2008


• Schoenfeld 1976

For the nth prime upper bound, Axler 2013 viii Korollar G is best when $n ge 8009824$. Dusart 2010 page 2 is best from $688383$ to $8009823$. Below that, either Dusart 2010 page 7, Dusart 1999 page 14, Robin 1983 or tweak the values since this is a limited range.

For the $n$-th prime lower bound, Axler 2013 vii Korollar I is best. For small values you can tweak things to get tighter.

By tweak, I mean something like starting with the Dusart 2010 formula $pi(x) le fracxln xbig(1 + frac1ln x + fracCln^2 xbig)$ where he proves $C=2.334$ works for $x ge 2953652287$. But we can take a piecewise range and come up with a value of $C$ that works for all values in the range, which can give a tighter result within that range. If you're not interested in small inputs where we can do this sort of computational testing, or you don't care about this sort of small optimization, then by all means just use one or more of the simpler formulas given in the papers.

If you want even tighter bounds, it turns out that the published bounds for prime counts are tighter than the nth prime bounds. By using an inverse lookup (binary search) of the opposite prime count bound, we can achieve tighter bounds for a pretty large range. In particular, Büthe 2014 gives a large range (to $1.4times 10^25$) where the Schoenfeld 1976 bounds apply irrespective of the truth of the Riemann Hypothesis, and Axler's bounds are also very good. At $10^17$ this gives a couple orders of magnitude tighter bound on both the upper and lower nth prime. If you want to assume the Riemann Hypothesis then of course you can continue using the Schoenfeld bounds.

share|cite|improve this answer

edited May 7 '15 at 6:24

answered May 7 '15 at 5:45

DanaJDanaJ

2,44211017

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you saying that Rosser/Schoenfeld bounds are conditional on RH?
  $endgroup$
  – Erick Wong
  May 7 '15 at 6:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Whoops, no the Rosser/Schoenfeld 1961 bounds do not. I was thinking of the Schoenfeld 1976 (ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0457374-X/…) bounds which do.
  $endgroup$
  – DanaJ
  May 7 '15 at 6:19
  
  

  
  
  
  

add a comment | 

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12

$begingroup$

See the papers of:

• Dusart 2010


• Axler 2013


• Axler 2014


• Büthe 2014


• Kotnick 2008


• Schoenfeld 1976

For the nth prime upper bound, Axler 2013 viii Korollar G is best when $n ge 8009824$. Dusart 2010 page 2 is best from $688383$ to $8009823$. Below that, either Dusart 2010 page 7, Dusart 1999 page 14, Robin 1983 or tweak the values since this is a limited range.

For the $n$-th prime lower bound, Axler 2013 vii Korollar I is best. For small values you can tweak things to get tighter.

By tweak, I mean something like starting with the Dusart 2010 formula $pi(x) le fracxln xbig(1 + frac1ln x + fracCln^2 xbig)$ where he proves $C=2.334$ works for $x ge 2953652287$. But we can take a piecewise range and come up with a value of $C$ that works for all values in the range, which can give a tighter result within that range. If you're not interested in small inputs where we can do this sort of computational testing, or you don't care about this sort of small optimization, then by all means just use one or more of the simpler formulas given in the papers.

If you want even tighter bounds, it turns out that the published bounds for prime counts are tighter than the nth prime bounds. By using an inverse lookup (binary search) of the opposite prime count bound, we can achieve tighter bounds for a pretty large range. In particular, Büthe 2014 gives a large range (to $1.4times 10^25$) where the Schoenfeld 1976 bounds apply irrespective of the truth of the Riemann Hypothesis, and Axler's bounds are also very good. At $10^17$ this gives a couple orders of magnitude tighter bound on both the upper and lower nth prime. If you want to assume the Riemann Hypothesis then of course you can continue using the Schoenfeld bounds.

share|cite|improve this answer

edited May 7 '15 at 6:24

answered May 7 '15 at 5:45

DanaJDanaJ

2,44211017

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you saying that Rosser/Schoenfeld bounds are conditional on RH?
  $endgroup$
  – Erick Wong
  May 7 '15 at 6:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Whoops, no the Rosser/Schoenfeld 1961 bounds do not. I was thinking of the Schoenfeld 1976 (ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0457374-X/…) bounds which do.
  $endgroup$
  – DanaJ
  May 7 '15 at 6:19
  
  

  
  
  
  

add a comment | 

12

$begingroup$

See the papers of:

• Dusart 2010


• Axler 2013


• Axler 2014


• Büthe 2014


• Kotnick 2008


• Schoenfeld 1976

For the nth prime upper bound, Axler 2013 viii Korollar G is best when $n ge 8009824$. Dusart 2010 page 2 is best from $688383$ to $8009823$. Below that, either Dusart 2010 page 7, Dusart 1999 page 14, Robin 1983 or tweak the values since this is a limited range.

For the $n$-th prime lower bound, Axler 2013 vii Korollar I is best. For small values you can tweak things to get tighter.

By tweak, I mean something like starting with the Dusart 2010 formula $pi(x) le fracxln xbig(1 + frac1ln x + fracCln^2 xbig)$ where he proves $C=2.334$ works for $x ge 2953652287$. But we can take a piecewise range and come up with a value of $C$ that works for all values in the range, which can give a tighter result within that range. If you're not interested in small inputs where we can do this sort of computational testing, or you don't care about this sort of small optimization, then by all means just use one or more of the simpler formulas given in the papers.

If you want even tighter bounds, it turns out that the published bounds for prime counts are tighter than the nth prime bounds. By using an inverse lookup (binary search) of the opposite prime count bound, we can achieve tighter bounds for a pretty large range. In particular, Büthe 2014 gives a large range (to $1.4times 10^25$) where the Schoenfeld 1976 bounds apply irrespective of the truth of the Riemann Hypothesis, and Axler's bounds are also very good. At $10^17$ this gives a couple orders of magnitude tighter bound on both the upper and lower nth prime. If you want to assume the Riemann Hypothesis then of course you can continue using the Schoenfeld bounds.

share|cite|improve this answer

edited May 7 '15 at 6:24

answered May 7 '15 at 5:45

DanaJDanaJ

2,44211017

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Are you saying that Rosser/Schoenfeld bounds are conditional on RH?
  $endgroup$
  – Erick Wong
  May 7 '15 at 6:07
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Whoops, no the Rosser/Schoenfeld 1961 bounds do not. I was thinking of the Schoenfeld 1976 (ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0457374-X/…) bounds which do.
  $endgroup$
  – DanaJ
  May 7 '15 at 6:19
  
  

  
  
  
  

add a comment | 

$begingroup$

See the papers of:

• Dusart 2010


• Axler 2013


• Axler 2014


• Büthe 2014


• Kotnick 2008


• Schoenfeld 1976

For the nth prime upper bound, Axler 2013 viii Korollar G is best when $n ge 8009824$. Dusart 2010 page 2 is best from $688383$ to $8009823$. Below that, either Dusart 2010 page 7, Dusart 1999 page 14, Robin 1983 or tweak the values since this is a limited range.

For the $n$-th prime lower bound, Axler 2013 vii Korollar I is best. For small values you can tweak things to get tighter.

By tweak, I mean something like starting with the Dusart 2010 formula $pi(x) le fracxln xbig(1 + frac1ln x + fracCln^2 xbig)$ where he proves $C=2.334$ works for $x ge 2953652287$. But we can take a piecewise range and come up with a value of $C$ that works for all values in the range, which can give a tighter result within that range. If you're not interested in small inputs where we can do this sort of computational testing, or you don't care about this sort of small optimization, then by all means just use one or more of the simpler formulas given in the papers.

If you want even tighter bounds, it turns out that the published bounds for prime counts are tighter than the nth prime bounds. By using an inverse lookup (binary search) of the opposite prime count bound, we can achieve tighter bounds for a pretty large range. In particular, Büthe 2014 gives a large range (to $1.4times 10^25$) where the Schoenfeld 1976 bounds apply irrespective of the truth of the Riemann Hypothesis, and Axler's bounds are also very good. At $10^17$ this gives a couple orders of magnitude tighter bound on both the upper and lower nth prime. If you want to assume the Riemann Hypothesis then of course you can continue using the Schoenfeld bounds.

share|cite|improve this answer

edited May 7 '15 at 6:24

answered May 7 '15 at 5:45

DanaJDanaJ

2,44211017

$endgroup$

share|cite|improve this answer

edited May 7 '15 at 6:24

answered May 7 '15 at 5:45

DanaJDanaJ

2,44211017

answered May 7 '15 at 5:45

DanaJDanaJ

2,44211017

answered May 7 '15 at 5:45

DanaJDanaJ

2,44211017