Prev: is Riemann Hypothesis implied by the Cramer conjecture? #692 Correcting Math
Next: the *ultimate equivalent* Riemann Hypothesis statement? #694 Correcting Math
From: Archimedes Plutonium on 17 Jul 2010 14:43
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 ---

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.

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.

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
 | 
Pages: 1
Prev: is Riemann Hypothesis implied by the Cramer conjecture? #692 Correcting Math
Next: the *ultimate equivalent* Riemann Hypothesis statement? #694 Correcting Math