- 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????

My advance thanks to you all

Lots of regards,

S.R.Sudarshan Iyengar

[Non-text portions of this message have been removed] - AFAIK, the simple relation between primes and the

Riemann Hypothesis is the formula:

oo

----- 1

Zeta(z)= | | ----------

| | 1-p^(-z)

p prime

Basically, the multiplication over all primes... etc.

I'm sure you are familiar with the formula.

Now, the above multiplication converging to zero for

any particular z (Im(z) not zero) is the whole point

of the hypothesis. In particular, notice that if

z=-2n+0i for n a natural number, the formula quickly

converges to zero. These are the trivial zeroes.

The nontrivial zeroes are expected to lie on the

complex line z=1/2+bi for some reals b. It is not

known which subset of the reals or what cardinality

they fall into.

The biggest thing about the 'distribution of primes'

is to dig into the complex algebra a little and find a

couple things like p^a and cos(b*ln(p)) running around

(z=a+bi). If the ln function maps most of the primes

into (for example) 2n*pi < ln(p) < pi+2n*pi, then

multiplying by a real number b might change that...

some thoughts

-jtpk

--- "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????

>

>

> My advance thanks to you all

>

>

> Lots of regards,

> S.R.Sudarshan Iyengar

>

>

> [Non-text portions of this message have been

> removed]

>

>

Do You Yahoo!?

Yahoo! Health - your guide to health and wellness

http://health.yahoo.com - --- "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 formula

for

> 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

Pages:

http://primepages.org/notes/rh.html

Google also found:

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

__________________________________________________

Do You Yahoo!?

Yahoo! Health - your guide to health and wellness

http://health.yahoo.com - 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

To: primenumbers

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 formula

for

> 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

Pages:

http://primepages.org/notes/rh.html

Google also found:

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

__________________________________________________

Do You Yahoo!?

Yahoo! Health - your guide to health and wellness

http://health.yahoo.com

Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com

The Prime Pages : http://www.primepages.org

Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/