Proof that there is a prime in ]m·n, (m+1)·n[
Xavier
Someone could help me? Given m > 0. Using Prime Number Theorem, I want
to prove that there exists n_0 such that for all n > n_0 there is a
prime in the interval ]m·n, (m+1)·n[. Specifically I want to know a
upper bound of n_0.
All of my tries are in vanuous. Is there any know upper bound for n_0?
Thanks a lot,
Xan.
PS: Paulo Ribenboim's book says that this assertion is true as a
consequence of a PNT. But obviously, there is no upper bound on it ;-)
Olivier
This is well-known, when you look at it in the proper way :)
Set X = mn and eps = 1/m. You want a prime in the interval
[X, (1+eps) X].
A usual roundabout is to look at
\vartheta(y)=\sum_{p\le y} \Log p
If you find X0 such that
\vartheta(y) = y (1+O^*(epsprime)) (where z=O^*(u) means |z|\le u)
then
\vartheta((1+eps)X)-\vartheta(X) = eps X + O^*( 2 epsprime X)
for X >= X0 and thus, if 2 epsprime < eps, then
\vartheta((1+eps)X)-\vartheta(X) > 0
which is equivalent to proving that the interval
]X, (1+eps) X] contains a prime and getting an explicit
value for X0. We have that with decent value of X0
roughly for eps >= 10^{-6}.
You can also try to work on \vartheta((1+eps)X)-\vartheta(X)
directly, see
http://math.univ-lille1.fr/~ramare/Maths/gap.pdf
References are below.
The main three are Rosser & Schoenfeld, Dusart and Ramare & Saouter.
Kadiri & student will soon get better results if my spy network is
correct :)
Good work!
Amities,
Olivier
So references:
@PhdThesis{Dusart*98,
author = {P. Dusart},
title = {Autour de la fonction qui compte le nombre de nombres
premiers},
school = {Limoges},
year = {1998},
OPTkey = {},
OPTtype = {},
address = {
\url{http\string://www.unilim.fr/laco/theses/1998/T1998_01.pdf}},
OPTmonth = {},
note = {173~pp},
OPTannote = {}
}
@Article{Dusart*99-1,
author = {P. Dusart},
title = {In\'egalit\'es explicites pour $\psi(X)$, $\theta(X)$,
$\pi(X)$ et les nombres premiers},
journal = {C. R. Math. Acad. Sci., Soc. R. Can.},
year = {1999},
OPTkey = {},
volume = {21},
number = {2},
OPTmonth = {},
pages = {53-59},
OPTnote = {},
OPTannote = {}
}
@Article{CostaPereira*89,
author = {N. {Costa Pereira}},
title = {Elementary estimates for the {C}hebyshev function
$\psi(X)$
and for the {M}\"obius function $M(X)$},
journal = {Acta Arith.},
year = {1989},
OPTkey = {},
volume = {52},
OPTnumber = {},
OPTmonth = {},
pages = {307-337},
OPTnote = {},
OPTannote = {}
}
@Article{Cramer*36,
author = {H. {Cramer}},
title = {On the order of magnitude of the difference between
consecutive prime numbers},
journal = {Acta Arith.},
year = {1936},
OPTkey = {},
volume = {2},
OPTnumber = {},
OPTmonth = {},
pages = {23-46},
OPTnote = {},
OPTannote = {}
}
@Article{Ramare-Saouter*02,
author = {O. {Ramar\'e} and Y. Saouter},
title = {Short effective intervals containing primes},
journal = {J. Number Theory},
year = {2003},
OPTkey = {},
volume = {98},
number = {},
OPTmonth = {},
pages = {10-33},
OPTnote = {},
OPTannote = {}
}
@Article{Rosser*41,
author = {J.B. {Rosser}},
title = {Explicit bounds for some functions of prime numbers},
journal = {American Journal of Math.},
year = {1941},
OPTkey = {},
volume = {63},
OPTnumber = {},
OPTmonth = {},
pages = {211-232},
OPTnote = {},
OPTannote = {}
}
@Article{Rosser-Schoenfeld*75,
author = { J.B. {Rosser} and L. {Schoenfeld}},
title = {Sharper bounds for the {C}hebyshev {F}unctions
$\vartheta(X)$
and $\psi(X)$},
journal = {Math. Comp.},
year = {1975},
OPTkey = {},
volume = {29},
number = {129},
OPTmonth = {},
pages = {243-269},
OPTnote = {},
OPTannote = {}
}