## TR: [PrimeNumbers] N-th prime boundary?

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• To PrimeNumbers groups ... De : Didier van der Straten [mailto:vdstrat@attglobal.net] Envoyé : jeudi 25 décembre 2003 22:17 À : Paul Leyland Objet : RE:
Message 1 of 3 , Dec 26, 2003

-----Message d'origine-----
De : Didier van der Straten [mailto:vdstrat@...]
Envoyé : jeudi 25 décembre 2003 22:17
À : Paul Leyland
Objet : RE: [PrimeNumbers] N-th prime boundary?

the "prime count function" up to N, while I believe the question is how to
approach
the Nth prime, with a value slightly larger than the actual Nth prime,
which would suggest a formula such as N * (ln N + k). Finding a "good" k
might be
another challenge, including checking it for high values of N.

Or I obviously missed one point. Forgive me... at Christmas or at New Year's
eve.
Didier

-----Message d'origine-----
De : Paul Leyland [mailto:pleyland@...]
Envoyé : mardi 23 décembre 2003 18:05
Objet : RE: [PrimeNumbers] N-th prime boundary?

I believe that N/(ln N - 2) will serve your purpose.

Paul

> -----Original Message-----
> From: William Bouris [mailto:melangebillyb@...]
> Sent: 23 December 2003 16:16
> Subject: [PrimeNumbers] N-th prime boundary?
>
> Hello, Group.
>
> Is there a simple formula for calculating an upper boundary
> which would always contain a particular nth-prime number?
> ie. input n --> formula ---> calculated limit > prime p. I
> bet that this question has been asked several times, but I
> just need a "slightly larger than p" value for any given n.
>
>
>
>
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• ... From: bobs@rsa.com Subject: Re: Upper bounds for the nth prime number Date: 1999/03/18 The Rosser & Schoenfeld results, as improved by Robin are the best
Message 2 of 3 , Dec 26, 2003
I found the following sci.math posting which may be of use:

--------------------------------------------------------------
From: bobs@...
Subject: Re: Upper bounds for the nth prime number
Date: 1999/03/18

The Rosser & Schoenfeld results, as improved by Robin are the best known.

p_n > n log n + n (log log n - 1.0072629)

for n > 7021, p_n <= nlog n + n (log log n - .9385)

-------------------------------------------------------------

I don't know a reference for this. Perhaps it can be found in the
following paper I found on MathSciNet.

Massias, Jean-Pierre(F-LIMOS-OC); Robin, Guy(F-LIMOS-OC)
Bornes effectives pour certaines fonctions concernant les nombres
premiers. (French. French summary) [Effective bounds for certain functions
involving primes] J. Thor. Nombres Bordeaux 8 (1996), no. 1, 215--242.
Abstract:
Let $p_k$ denote the $k$th prime number and let
$\theta(p_k)=\sum^k_{i=1}\log p_i$ be the Chebyshev function. The authors
obtain new explicit upper and lower bounds for the expressions $p_k$,
$\theta(p_k)$, $\sum^k_{i=1}p_i$ and $\sum_{p\leq x}p$, thereby improving
several estimates obtained by J. B. Rosser and L. Schoenfeld\
\ref[Illinois J. Math. 6 (1962), 64--94; MR 25 #1139; Math. Comp. 29
(1975), 243--269; MR 56 #15581a] and Robin \ref[Acta Arith. 42 (1983),
no. 4, 367--389; MR 85j:11109].
----------------------------------------------------------------

---Edwin

------------------------------------------------------------
W. Edwin Clark, Math Dept, University of South Florida,
http://www.math.usf.edu/~eclark/
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• ... I m not quite sure what you re asking. Are you asking for an upper bound on the nth prime number? A number that is slightly larger than the nth prime?
Message 3 of 3 , Dec 26, 2003
At 05:46 AM 12/26/2003, Didier van der Straten wrote: