New formula for approximating pi(n)
- Hey guys,
a few weeks ago I made a post in which I showed a proof for the
formula: pi(n) ~ n( .5 -sqrt(.25 - 1/ln(n))) and also showed that it
was a better approximation to pi(n) over n/ln(n) and n/(ln(n)-1).
In my paper (posted at http://rusi.greatnow.com/Math/ApproximatingPi
(n).pdf) I proved a few other formulae. Last week I realized that I
can actually use them to improve (or at least try to) over Li(n) and
R(n) - the Riemann Function.
So far I have the formula
K(n) = Li(n) / ( 1 - sum( mobius(k)/n^((k-1)/k), k=2..infinity) )
and of course K(n) ~ Pi(n).
I can prove that it is better than Li(n) and I have tested and it is
better than R(n) for at least half the cases I tried. The problem is
that I do not have the computational power to test this formula for
large values of k. The largest I can afford is k(max)=10^5.
Well I have made a table that you can view at:
Feel free to e-mail me at rusi_kolev@... to tell me what you
P.S. oh and if for some reason you can not open, you can access
to view the formula at least ....