Mixing
Upper and Lower bounds on the nth prime number
Clash Royale CLAN TAG#URR8PPP
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I know from papers like Dusarts's that $$nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +122 ln^2 nright) leq p_n leq nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +102 ln^2 nright) $$
But could any one give and explicit number $n_0$ for which for all $n geq n_0$ the above inequalities hold true ?
It does not have to be the smallest one, also if there is way, could you please show me how to get this number
I read a paper that estimate that the $n_0 approx 3.9*10^31$ under the R.H.
if we don't assume R.H. could we still have $n_0 ll 10^100$
Thanks in advance
number-theory prime-numbers
edited Aug 2 at 20:49
Ed Pegg
9,12432486
asked Aug 2 at 17:57
Ahmad
2,2911625
This question has an open bounty worth +500
reputation from Ahmad ending ending at 2018-08-12 07:00:38Z">in 4 days.
This question has not received enough attention.
Give a ref to a paper dealing with this problem, or give a proof about the $n_0$ number, Thanks
add a comment |Â
up vote
2
down vote
favorite
I know from papers like Dusarts's that $$nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +122 ln^2 nright) leq p_n leq nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +102 ln^2 nright) $$
But could any one give and explicit number $n_0$ for which for all $n geq n_0$ the above inequalities hold true ?
It does not have to be the smallest one, also if there is way, could you please show me how to get this number
I read a paper that estimate that the $n_0 approx 3.9*10^31$ under the R.H.
if we don't assume R.H. could we still have $n_0 ll 10^100$
Thanks in advance
number-theory prime-numbers
edited Aug 2 at 20:49
Ed Pegg
9,12432486
asked Aug 2 at 17:57
Ahmad
2,2911625
This question has an open bounty worth +500
reputation from Ahmad ending ending at 2018-08-12 07:00:38Z">in 4 days.
This question has not received enough attention.
Give a ref to a paper dealing with this problem, or give a proof about the $n_0$ number, Thanks
These bounds are much tighter than the one mentioned in Wikipedia! Did the paper mentioned a proven bound (not assuming RH) ?
– Peter
Aug 2 at 22:06
My guess is that $n_0=10^100$ is sufficient
– Peter
Aug 2 at 22:09
@peter yes,search for juan and toulisse paper
– Ahmad
Aug 2 at 23:35
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I know from papers like Dusarts's that $$nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +122 ln^2 nright) leq p_n leq nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +102 ln^2 nright) $$
But could any one give and explicit number $n_0$ for which for all $n geq n_0$ the above inequalities hold true ?
It does not have to be the smallest one, also if there is way, could you please show me how to get this number
I read a paper that estimate that the $n_0 approx 3.9*10^31$ under the R.H.
if we don't assume R.H. could we still have $n_0 ll 10^100$
Thanks in advance
number-theory prime-numbers
edited Aug 2 at 20:49
Ed Pegg
9,12432486
asked Aug 2 at 17:57
Ahmad
2,2911625
I know from papers like Dusarts's that $$nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +122 ln^2 nright) leq p_n leq nleft(ln n +ln ln n -1+ fracln ln n -2ln n-fracln^2 ln n -6 ln ln n +102 ln^2 nright) $$
But could any one give and explicit number $n_0$ for which for all $n geq n_0$ the above inequalities hold true ?
It does not have to be the smallest one, also if there is way, could you please show me how to get this number
I read a paper that estimate that the $n_0 approx 3.9*10^31$ under the R.H.
if we don't assume R.H. could we still have $n_0 ll 10^100$
Thanks in advance
number-theory prime-numbers
edited Aug 2 at 20:49
Ed Pegg
9,12432486
asked Aug 2 at 17:57
Ahmad
2,2911625
edited Aug 2 at 20:49
Ed Pegg
9,12432486
edited Aug 2 at 20:49
Ed Pegg
9,12432486
edited Aug 2 at 20:49
Ed Pegg
9,12432486
9,12432486
asked Aug 2 at 17:57
Ahmad
2,2911625
asked Aug 2 at 17:57
Ahmad
2,2911625
asked Aug 2 at 17:57
Ahmad
2,2911625
2,2911625
This question has an open bounty worth +500
reputation from Ahmad ending ending at 2018-08-12 07:00:38Z">in 4 days.
This question has not received enough attention.
Give a ref to a paper dealing with this problem, or give a proof about the $n_0$ number, Thanks
This question has an open bounty worth +500
reputation from Ahmad ending ending at 2018-08-12 07:00:38Z">in 4 days.
This question has not received enough attention.
Give a ref to a paper dealing with this problem, or give a proof about the $n_0$ number, Thanks
These bounds are much tighter than the one mentioned in Wikipedia! Did the paper mentioned a proven bound (not assuming RH) ?
– Peter
Aug 2 at 22:06
My guess is that $n_0=10^100$ is sufficient
– Peter
Aug 2 at 22:09
@peter yes,search for juan and toulisse paper
– Ahmad
Aug 2 at 23:35
add a comment |Â
These bounds are much tighter than the one mentioned in Wikipedia! Did the paper mentioned a proven bound (not assuming RH) ?
– Peter
Aug 2 at 22:06
My guess is that $n_0=10^100$ is sufficient
– Peter
Aug 2 at 22:09
@peter yes,search for juan and toulisse paper
– Ahmad
Aug 2 at 23:35
These bounds are much tighter than the one mentioned in Wikipedia! Did the paper mentioned a proven bound (not assuming RH) ?
– Peter
Aug 2 at 22:06
These bounds are much tighter than the one mentioned in Wikipedia! Did the paper mentioned a proven bound (not assuming RH) ?
– Peter
Aug 2 at 22:06
My guess is that $n_0=10^100$ is sufficient
– Peter
Aug 2 at 22:09
My guess is that $n_0=10^100$ is sufficient
– Peter
Aug 2 at 22:09
@peter yes,search for juan and toulisse paper
– Ahmad
Aug 2 at 23:35
@peter yes,search for juan and toulisse paper
– Ahmad
Aug 2 at 23:35
add a comment |Â
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These bounds are much tighter than the one mentioned in Wikipedia! Did the paper mentioned a proven bound (not assuming RH) ?
– Peter
Aug 2 at 22:06
My guess is that $n_0=10^100$ is sufficient
– Peter
Aug 2 at 22:09
@peter yes,search for juan and toulisse paper
– Ahmad
Aug 2 at 23:35