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• The big thing with RH is in accurately defining the error term between pi(x), the count of primes to x, and Li(x), the best known approximation, defined by
Message 1 of 6 , May 6, 2002
The big thing with RH is in accurately defining the error term between
pi(x), the count of primes to x, and Li(x), the best known approximation,
defined by integral(t=0,x, of 1/logt).

logt is lnt, i.e. the natural logarithm, in base e.

If RH is true, then we can say that pi(x) ~ Li(x).

More accurately, if RH is true then we have:

pi(x) = Li(x) + O(sqrt(x).logx)

which is a far better error bound that currently known.

Other connections between RH and the primes include:

1/zeta(p) is asymtopic to the count of p-free numbers, e.g. if p=5, then
1/zeta(5) is asymtopic to the count of numbers not involving some q^5.
(see: http://mathworld.wolfram.com/RiemannZetaFunction.html)

If psi(x) is the count of primes and prime powers less than x, then the
zeroes of Riemann's Zeta function can be used to determine psi(x). (almost -
psi(x) is discontinuous, so RZF defines a modification of psi(x), namely
psi0(x).
(see:
http://users.globalnet.co.uk/~perry/maths/riemannshypothesis/riemannshypothe
sis.htm)

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Phil Carmody [mailto:thefatphil@...]
Sent: 06 May 2002 07:44

--- "S.R.Sudarshan Iyengar" <gayathrisr@...> wrote:
> Dear members,
>
> I recently heard that the application of Riemann
hypothesis
> will give a hint on the distribution of primes (also, a formula
for
> the nth prime can be obtained)???
>
> But I am unable to see any relation between Riemann
> hypothesis and Prime numbers. Can anyone help me in this
regard????

First hit from the search string 'Riemann Hypothesis' on the Prime
Pages:
http://primepages.org/notes/rh.html

http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html

Phil

=====
--
"One cannot delete the Web browser from KDE without
losing the ability to manage files on the user's own
hard disk." - Prof. Stuart E Madnick, MIT.
So called "expert" witness for Microsoft. 2002/05/02

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