Problems & Puzzles: Conjectures

Conjecture 67. primes & e

Anton Vrba sent the following conjecture:

The limit of the ratio p to the π(p)th root of the primorial p# is equal to e, where p is prime approaching infinity, π() being the prime counting function and e=2.71828… is the  base of the natural logarithm.

Lim as p ->∞ {p/(p#)^(1/π(p))} = e ... (1)

Q1:  Is equation (1) a re-invention?

Q2:  Can you prove or disprove for (1)?

Contribution came from Antoine Verroken

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Antoine wrote:

I think there is some error in the formula :

- p# = product of all primes up to p
- the theta-function of Chebychev  : theta ( p ) = sum ( ln ( p ) for all primes =< p) is asymptotically equal to p ->  p# ~ e ^ p
- n(p) : number of primes up to p = ~ p / ln ( p )
- then  p / p# ^ ( 1 / n(p) ) becomes p / ( e ^ p ) ^ ( ln ( p ) / p = 1

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Werner Sand wrote:

Analogy from the natural numbers:

n / [n! ^ (1/n)]  =~  n / { [ sqrt (2 π n) n^n  e^-n]  ^ (1/n) }    (Stirling)
=   n [ (e^n) / sqrt (2 π n) n^n ] ^ (1/n)  =   e / [ sqrt (2 π n)  ^ (1/n)].
lim (n->inf) … =  e       (since  lim  [sqrt (2 π n)]  ^ (1/n)  = 1)

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Emmanuel Vantieghem wrote:

I transformed the conjecture in logarithmic form :

Limit(n->infinity)[ log p_n – ( log p_n#)/n ] =?= 1

Numerically, the value between [ ] is about 1.0756 when n = 10^6 and about 1.063541 when n = 10^7. So, if there is convergence, it’s very slowly. But it still remains possible that the conjecture is true. However, I think a proof will use more knowledge than the PNT (maybe the Riemann Hypothesis should become settled first).

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Alexei Kourbatov wrote on Feb. 24, 2016:

Conjecture 67 is true. However the convergence to e is very slow:

p(n)/(p(n)#)^(1/n) is approximately exp(1 + 1/ln p(n) + 3/ln^2 p(n) ).

Here is a link to my proof:
http://www.javascripter.net/math/publications/AKourbatov_GeoMean_of_n_primes.pdf

...
Even more surprisingly, it turned out that Conjecture 67 has been proved in 2011! I learned that only an hour ago, from a more thorough Web search. Here is the relevant paper, by J. Sandor and A. Verroken:

My new paper is still a step forward - I prove for the first time the formula exp(1 + 1/ln p(n) + 3/ln^2 p(n) ) and give (without proof) a family of potentially better formulas (for large n); see remark (ii). Anyway, now we have multiple proofs, that's a good thing.

I updated my paper, to refer to the 2011 proof as well as a further development of 2012. The link is the same.

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