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From: Archimedes Plutonium on 17 Jul 2010 15:41


Archimedes Plutonium wrote:
> I am looking for the best Riemann Hypothesis equivalent statement to
> tie in the Indirect Euclid Infinitude of Primes proof method. By
> correcting that flaw of logic that both P-1
> and P+1 are necessarily prime, yielding the infinitude of Twin Primes,
> I suspect is a
> key to proving the Riemann Hypothesis RH.
>
> So I looked for equivalent RH statements:
> --- quoting Wikipedia in part ---
> Riemann's explicit formula for the number of primes less than a given
> number in terms of a sum over the zeros of the Riemann zeta function
> says that the magnitude of the oscillations of primes around their
> expected position is controlled by the real parts of the zeros of the
> zeta function. In particular the error term in the prime number
> theorem is closely related to the position of the zeros: for example,
> the supremum of real parts of the zeros is the infimum of numbers β
> such that the error is O(xβ) (Ingham 1932).
>
>
> Von Koch (1901) proved that the Riemann hypothesis is equivalent to
> the "best possible" bound for the error of the prime number theorem.
>
>
> A precise version of Koch's result, due to Schoenfeld (1976), says
> that the Riemann hypothesis is equivalent to. . .
>
> --- end quoting ---

Ingham, Von Koch, and Schoenfeld and others bespeak of the Riemann
Hypothesis
as the most efficient placing of primes in a prime distribution. As if
efficiency and accuracy
of placement of primes is what the Riemann Hypothesis is all about.


>
> Let me try to give an equivalent RH statement myself.
>
> It is already proven, I think it was Chebychev, that between n and 2n
> always exists another prime.
>
> So, let me focus on n+1 and 2n-1
>
> We have:
>
> for 2, 2+1 = 3 and 4-1 = 3
>
> for 3, 3+1=4 and 6-1=5
>
> for 4, 4+1 =5 and 8-1=7
>
> for 5, 5+1=6 and 10-1=9
>
> etc etc
>
> Now, instead of Riemann getting involved with the Complex Number
> Plane, how about a
> Riemann Hypothesis more down to Earth. How about a Riemann Hypothesis
> with just the plain old Natural Numbers since we find billions and
> zillions of equivalent statements, but
> never the most simple statement.
>
> So let me proffer my own equivalent statement of the Riemann
> Hypothesis since the one
> thing that RH can never get away from is the distribution of prime
> numbers.
>
> Archimedes Plutonium's equivalent statement of the Riemann Hypothesis:
> The RH, if true says that as n becomes large, very large that both n+1
> and 2n-1
> are both prime numbers. If that is true, then a proof of that RH
> equivalent is easily
> begot from the Euclid Infinitude of Primes proof Indirect method for
> it makes
> n+1 and 2n-1 necessarily new primes as n goes to infinity.


Then this equivalent statement to the RH by myself is not efficient
and accurate enough.

I should have said that the RH equivalent is such that n-1, n+1 and
2n-1, 2n+1, all four
of those numbers are necessarily prime as n tends to infinity.

An example of that is n=30 so that n-1 =29 and n+1=31, and
2n-1=60--1=59 and 2n+1=
60+1=61 are all four prime numbers. So that would be a Maximum density
of primes
given n goes to infinity.

It is where the Infinitude of Primes proof conjoins with the Riemann
Hypothesis, and the
proof of this RH is simply a Indirect Method with Mathematical
Induction that yields four
Euclid Numbers, all four of which are necessarily prime numbers.



>
> Now I am curious since I define with precision the finite-number
> versus the infinite-number
> as the boundary at 10^500. So I am curious as to whether 10^500 (+1)
> is a prime number
> and its associate of 2x(10^500) -1. If not, then let us chose as the
> boundary where n+1
> and 2n-1 in the region of 10^500 are both prime numbers. So that
> mathematics does share
> a input into the selection of the boundary between finite and infinite-
> number.
>
> Perhaps a major reason the RH was never proven or steered into a
> correct path to prove it, was that it was too much cloaked in the
> Complex Number Plane and if someone had retrieved it out of that
> cloaking, would have seen it in its more basic form that n+1 and 2n-1
> are both
> primes when n tends to infinity. They may not have realized that a
> simple tinker to fix the logic flaw of Euclid IP indirect, but at
> least they would have made RH more understandable.
> Mathematicians are like artists, once they paint legs on a snake, they
> refuse to remove the legs and rather increase the complexity.
>

Now Physics is the king of sciences and mathematics is only a room, a
tiny
room in the house of physics. And Physics would define the boundary
between
finite number versus infinite-number and it would be the largest
Planck unit
which is the Coulomb Interactions in element 100 of about 10^500. But
here is where
mathematics has a "say at the table". Since the RH of above would have
four primes
at n, 2n, the question is does 10^500 plus and minus 1 yield twin
primes and does
2x10^500 plus and minus 1 yield twin primes? If so, then we assuredly
take 10^500 as
the boundary between finite-number versus infinite-number. Or if there
is another large
number in the vicinity of 10^500 that yields those four primes.

Carbon in me, carbon of plutonium, fill me with life anew, that I may
love what thou dost
love. Oxygen in me, oxygen of plutonium. .

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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