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
***
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
***
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)
***
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, its 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).
***
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.
***
