Are there better explicit bounds on the nth prime than Rosser's?

Dror Speiser

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Jun 15, 2004, 10:30:02 PM6/15/04

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Hi,

I'm looking for good bounds on the nth prime, and the theta function
(log(procut(primes))). I looked ar Rosser's article, which gives good
stuff, but since it's from 1941, there might be some improvements...
So if you know an article or the actual bounds for these two wonderful
functions, please do share them with me.

What I really am looking for is an upper bound for the theta function
below e^95, which Rosser doesn't give (he does give something, but it
has a restriction that isn't good for me...). The lower bound he gives
is great, and so are the those for the nth prime.

Cheers,

Dror Speiser

Hugo Pfoertner

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Jun 24, 2004, 1:00:40 PM6/24/04

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Although I didn't read those articles, I guess that the bounds given in
Rosser's 1941 paper

J. B. Rosser, Explicit bounds for some functions of prime numbers,
Amer. J. Math. 63 (1941), 211--232.

have been improved in

J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev
functions Theta(X) and Psi(X) , Math. Comp. 29 (1975), 243--269.
and
L. Schoenfeld, Sharper bounds for the Chebyshev functions Theta(X)
and Psi(X). II, Math. Comp. 30 (1976), 337--360

(see http://www.ams.org/mcom/1996-65-213/S0025-5718-96-00669-2/home.html
)

The best? currently known lower bound for P_n might be

Dusart, P. "The k_th Prime is Greater than k(ln k + ln ln k - 1) for k
>= 2." Math. Comput. 68, 411-415, 1999.
(see http://mathworld.wolfram.com/RossersTheorem.html )

Hugo Pfoertner

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Jun 25, 2004, 2:31:15 PM6/25/04

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Hugo Pfoertner schrieb:

More information on this problem was given by Peter Luschny and Hermann
Kremer in a thread in the (German) newsgroup de.sci.mathematik, see
http://groups.google.com/groups?selm=cat037%24bmv%241%40online.de

Peter Luschny noted:
"Die obere Schranke p_k <= k(ln(k)+ln(ln(k)))
findet sich bei Rosser und Schoenfeld, der Beweis scheint
aber nicht einwandfrei zu sein, ist aber spätestens
seit Massias und Robin gesichert.
"
freely translated
"An upper bound p_k <= k(ln(k)+ln(ln(k)))
is found in the Rosser and Schoenfeld article, with a debatable proof.
The remaining doubts were resolved by Massias and Robin.
( J.-P. MASSIAS & G. ROBIN, Bornes effectives pour certaines
fonctions concernant les nombres premiers, Journal de Théorie
des Nombres de Bordeaux 8 (1996), 215-242. MR 97g:11099 )

Hugo

Alan Simpson

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Jun 28, 2004, 10:32:51 AM6/28/04

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hi,

just took a look at Massias and Robin's paper.

They have the bound
p_k \leq k(ln k + ln ln k - 1 + (ln ln k-1.8)/ln k).

Furthermore, I just did some computing and it seems that the 1.8 can
be replaced by something like 2.04 for k sufficiently large (note that
is just an observation, no claims to have proofs here).

Anyone have an idea what this term should really be?

Maybe 2+o(1)?

Can we derive it from the Prime Number theorem? Not the explicit upper
bound itself, but maybe an asymptotic version?

What about the next term? What should that look like?

Thanks!

Hugo Pfoertner <not...@abouthugo.de> wrote in message news:<cbhr1j$tk7$1...@news.ks.uiuc.edu>...