Pro Answers > Claymath Millennium Problems > The Riemann Hypothesis
Discuss The Riemann Hypothseis
The Riemann Hypothesis
Introduction
This is a conjecture about the distribution and frequency of prime numbers among natural numbers.
Prime
The Riemann hypothesis concerns prime numbers, which are numbers that can only be evenly divided by themselves and 1.
2, 3, 5, 7...
Distribution
The complete and accurate prediction of the frequency and distribution of prime numbers has so far defied accurate explanation.
Definite
The primes appear to follow no definite and regular distributional order.
Massive
The size of the group of all possible primes is so potentially massive that it seems as if we may never be able to solve the problem adequately.
Top of Page
Framework
We face the seemingly impossible task of completing a distributional framework of all prime numbers among all natural numbers, even though any solution may be infinite in size.
Pattern
Riemann discovered that the distribution of some special zeros seemed to follow a pattern, relative to a zeta function.
Conjecture
Riemann proposed a conjecture about certain zeros which seem to have a real part that lies at the point between between 0 and 1.
Real
This basically means that the real part of the nonobvious zeros is 1/2, though theoretically this value may be subject to fluctuation.
Top of Page
Interesting
Riemann's conjecture relates to interesting solutions of the equation ζ(s) = 0 which are thought to lie on a certain straight vertical line in the complex plane.
Critical
This means that the zeros all seem to align themselves on the critical line, with real part between zero and one (1/2).
The critical line in the complex plane.
Position
The diagram above shows a line representing the position of these zeros as studied using Riemann's zeta function.
Hypothesis
The Riemann Hypothesis states that all zeros "very probably" lie on this critical line, subject to the behaviour of the associated zeta function when it is equal to zero.
Top of Page
Contrary
A contrary instance, such as a single zero not on the critical line, disproves the singularly stated Riemann hypothesis.
Singularly
Riemann's conjecture still stands as singularly true, until proven otherwise.
Checked
The hypothesis has been checked for the first 1,500,000,000 of its solutions, but we still don't know for certain whether all zeros lie on the critical line unchangingly and forever.
Proof
We need to find an ultimate and singular proof, and to prove whether or not there will ever be a single contrary instance.
Summary
Do all Riemann zeta function zeros unchangingly lie on the critical line?
Are there unchanging patterns in the distribution of the primes?
The Millennium Problem
The Millennium problem is to prove that the Riemann Hypothesis is either true or false.
For the exact problem description please refer to Claymath.org
The Answer
Conjecture
The Riemann hypothesis presents us with two options, either the zeros all lie unchangingly on the critical line or they do not, and asks us to choose one of them.
Top of Page
Structure
To properly understand numbers and their distributional structure, we need to go back to their birth.
Numbers
Numbers are a human invention.
Symbolism
All numbers symbolise either one, more (or less) than one, or nothing (zero).
Repetition
Any larger combination of numbers is simply the same original three potentials being infinitely repeated by us.
Top of Page
Manipulation
These new amounts (numbers) are then manipulated to form other combinations but no matter how far we go with this process, we are still ultimately repeating these three original potentials.
Over and over and over again.
Endless
There is no unchanging beginning or end to numbers, there is a loop of endless repetition, invented and perpetuated by us.
Half
The Riemann zeta function zeros seem to lie on the critical line with real part between zero and one if the zeta function is equal to zero.
ζ(s) = 0
Nothing
Real part 1/2 symbolises nothing (neutral), and this is the key to understanding the Riemann hypothesis properly.
Top of Page
Between
The Riemann zeros appear at the neutral point between the two numbers (zero and one).
The critical line in the complex plane.
Neutral
The critical line is neutral in its position and the prime zeros appear to lie on a neutral line because real part 1/2 equates to a neutral potential.
Exhibit
It is because zeros with real part 1/2 are neutral in potential that they also exhibit neutral distributional characteristics when studied in sequence.
Inaccurate
The idea that we could singularly and unchangingly predict the complete distribution of the primes is inaccurate because it is singular and therefore it is missing its other two possible potentials.
Top of Page
Assume
The zeros seem to lie on the critical line with real part 1/2 so far, but to assume that this fact can never change is incorrect.
Opposite
I can simply state the opposite and neutral potentials to this singular statement and create a loop which disproves as singular (loops) the original assumption.
Fluctuations
There will always be the theoretical potential for fluctuations (changes) within any solution and the only way to avoid this is to realise that everything, regardless of relative detail, possesses the potential for three simultaneous actions.
Is the Riemann hypothesis true?
1. The Riemann hypothesis is true.
2. The Riemann hypothesis is false.
3. The Riemann hypothesis is neutral.
Simultaneously.
Am I wrong?
I simultaneously oppose, agree with, and neutralise all criticism ad infinitum.
My point is literal.
There is no point creating a theory of everything that doesn't work.
NEXT > Yang-Mills Theory and the Mass Gap Hypothesis
Interested?
If you're interested in a positive discussion about everything the whole of protheory.com is mirrored on my Theory Of Everything Forum
Discuss The Riemann Hypothseis