consecutive prime gaps and explicit bound
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I am aware of the theorem that $p_n+1- p_n leq n^0.525$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" explicit, is it known that $p_n+1-p_n leq c n^alpha$ for all $n geq 1$ and for small $c$, lets say $c leq 2$ and $alpha leq 0.55$ ?
Any ref that can give me the explicit numbers or a way to construct them would be great.
Thank you, also i posted the question yesterday on MSE
nt.number-theory reference-request analytic-number-theory prime-numbers prime-gaps
edited yesterday
GH from MO
56.4k5138214
asked yesterday
Ahmad
35019
add a comment |Â
up vote
5
down vote
favorite
I am aware of the theorem that $p_n+1- p_n leq n^0.525$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" explicit, is it known that $p_n+1-p_n leq c n^alpha$ for all $n geq 1$ and for small $c$, lets say $c leq 2$ and $alpha leq 0.55$ ?
Any ref that can give me the explicit numbers or a way to construct them would be great.
Thank you, also i posted the question yesterday on MSE
nt.number-theory reference-request analytic-number-theory prime-numbers prime-gaps
edited yesterday
GH from MO
56.4k5138214
asked yesterday
Ahmad
35019
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I am aware of the theorem that $p_n+1- p_n leq n^0.525$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" explicit, is it known that $p_n+1-p_n leq c n^alpha$ for all $n geq 1$ and for small $c$, lets say $c leq 2$ and $alpha leq 0.55$ ?
Any ref that can give me the explicit numbers or a way to construct them would be great.
Thank you, also i posted the question yesterday on MSE
nt.number-theory reference-request analytic-number-theory prime-numbers prime-gaps
edited yesterday
GH from MO
56.4k5138214
asked yesterday
Ahmad
35019
I am aware of the theorem that $p_n+1- p_n leq n^0.525$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" explicit, is it known that $p_n+1-p_n leq c n^alpha$ for all $n geq 1$ and for small $c$, lets say $c leq 2$ and $alpha leq 0.55$ ?
Any ref that can give me the explicit numbers or a way to construct them would be great.
Thank you, also i posted the question yesterday on MSE
nt.number-theory reference-request analytic-number-theory prime-numbers prime-gaps
nt.number-theory reference-request analytic-number-theory prime-numbers prime-gaps
edited yesterday
GH from MO
56.4k5138214
asked yesterday
Ahmad
35019
edited yesterday
GH from MO
56.4k5138214
asked yesterday
Ahmad
35019
edited yesterday
GH from MO
56.4k5138214
edited yesterday
GH from MO
56.4k5138214
edited yesterday
GH from MO
56.4k5138214
56.4k5138214
asked yesterday
Ahmad
35019
asked yesterday
Ahmad
35019
asked yesterday
Ahmad
35019
35019
add a comment |Â
add a comment |Â
1 Answer
1
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oldest
votes
up vote
8
down vote
accepted
The result you quote is due to Baker-Harman-Pintz (2000). I am not aware of any concrete effective version of this result, but if you increase the exponent $0.525$ to $2/3$, then such a variant is available by the work of Dudek. See also my response to this MO question.
answered yesterday
GH from MO
56.4k5138214
Thx, so is there $alpha <1$ such that $cleq 2$ and $n_0$ or $x_0$ is small say less than $10^10$ or $10^12 $ ?
– Ahmad
yesterday
@Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_n+1-p_n<n^3/4$. It is not clear to me what can be said for smaller $n$'s unconditionally.
– GH from MO
yesterday
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
8
down vote
accepted
The result you quote is due to Baker-Harman-Pintz (2000). I am not aware of any concrete effective version of this result, but if you increase the exponent $0.525$ to $2/3$, then such a variant is available by the work of Dudek. See also my response to this MO question.
answered yesterday
GH from MO
56.4k5138214
Thx, so is there $alpha <1$ such that $cleq 2$ and $n_0$ or $x_0$ is small say less than $10^10$ or $10^12 $ ?
– Ahmad
yesterday
@Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_n+1-p_n<n^3/4$. It is not clear to me what can be said for smaller $n$'s unconditionally.
– GH from MO
yesterday
add a comment |Â
up vote
8
down vote
accepted
The result you quote is due to Baker-Harman-Pintz (2000). I am not aware of any concrete effective version of this result, but if you increase the exponent $0.525$ to $2/3$, then such a variant is available by the work of Dudek. See also my response to this MO question.
answered yesterday
GH from MO
56.4k5138214
Thx, so is there $alpha <1$ such that $cleq 2$ and $n_0$ or $x_0$ is small say less than $10^10$ or $10^12 $ ?
– Ahmad
yesterday
@Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_n+1-p_n<n^3/4$. It is not clear to me what can be said for smaller $n$'s unconditionally.
– GH from MO
yesterday
add a comment |Â
up vote
8
down vote
accepted
up vote
8
down vote
accepted
The result you quote is due to Baker-Harman-Pintz (2000). I am not aware of any concrete effective version of this result, but if you increase the exponent $0.525$ to $2/3$, then such a variant is available by the work of Dudek. See also my response to this MO question.
answered yesterday
GH from MO
56.4k5138214
The result you quote is due to Baker-Harman-Pintz (2000). I am not aware of any concrete effective version of this result, but if you increase the exponent $0.525$ to $2/3$, then such a variant is available by the work of Dudek. See also my response to this MO question.
answered yesterday
GH from MO
56.4k5138214
edited yesterday
answered yesterday
GH from MO
56.4k5138214
answered yesterday
GH from MO
56.4k5138214
answered yesterday
GH from MO
56.4k5138214
56.4k5138214
Thx, so is there $alpha <1$ such that $cleq 2$ and $n_0$ or $x_0$ is small say less than $10^10$ or $10^12 $ ?
– Ahmad
yesterday
@Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_n+1-p_n<n^3/4$. It is not clear to me what can be said for smaller $n$'s unconditionally.
– GH from MO
yesterday
add a comment |Â
Thx, so is there $alpha <1$ such that $cleq 2$ and $n_0$ or $x_0$ is small say less than $10^10$ or $10^12 $ ?
– Ahmad
yesterday
@Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_n+1-p_n<n^3/4$. It is not clear to me what can be said for smaller $n$'s unconditionally.
– GH from MO
yesterday
Thx, so is there $alpha <1$ such that $cleq 2$ and $n_0$ or $x_0$ is small say less than $10^10$ or $10^12 $ ?
– Ahmad
yesterday
Thx, so is there $alpha <1$ such that $cleq 2$ and $n_0$ or $x_0$ is small say less than $10^10$ or $10^12 $ ?
– Ahmad
yesterday
@Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_n+1-p_n<n^3/4$. It is not clear to me what can be said for smaller $n$'s unconditionally.
– GH from MO
yesterday
@Ahmad: Dudek's result implies that if $n$ is very-very large (doubly exponentially large), then $p_n+1-p_n<n^3/4$. It is not clear to me what can be said for smaller $n$'s unconditionally.
– GH from MO
yesterday
add a comment |Â
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