RE: [PrimeNumbers] Riemann Hypothesis
- 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.
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
From: Phil Carmody [mailto:thefatphil@...]
Sent: 06 May 2002 07:44
Subject: Re: [PrimeNumbers] Riemann Hypothesis
--- "S.R.Sudarshan Iyengar" <gayathrisr@...> wrote:
> Dear members,hypothesis
> I recently heard that the application of Riemann
> will give a hint on the distribution of primes (also, a formulafor
> the nth prime can be obtained)???regard????
> But I am unable to see any relation between Riemann
> hypothesis and Prime numbers. Can anyone help me in this
First hit from the search string 'Riemann Hypothesis' on the Prime
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