The Riemann Hypothesis

[Pro]

The Riemann Hypothesis.

[Pro]

It's pretty much impossible for me to rewrite the protheory.com topics in full as I'd end up just making it too complicated so for now please just take a look at this original archive from my website and then you can ask me any questions below here

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.



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.



Interesting

Riemann's conjecture relates to interesting solutions of the equation æ(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.



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.



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.



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.

æ(s) = 0



Nothing

Real part 1/2 symbolises nothing (neutral), and this is the key to understanding the Riemann hypothesis properly.



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.



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.

[Pro]

The Riemann Hypothesis - Every single response I've ever had (completely unedited).

Pro - Administrator and Forum Founder



Still searching for falsity content within Pro Theory

[Pro]

My first ever exposure to the RH was in a New Scientist article entitled (I think) "Does nature dance to music from the primes?" I read this article back in about 2001 or so and this got me into this problem and the nature and relationship of the primes to the sequences of nature and the universe.

I realised that if I could explain the primes and why they exist fully I could by default also explain the RH as the whole problem is simply about whether or not prime numbers follow a certain unchanging pattern. Namely do all Riemann zeta function 'zeros' lie on the critical line ad infinitum?

Let's begin by saying that prime numbers as I understand them are numbers that may be evenly divided only by themselves and 1. So this is the basic definition of our building blocks, I say this because the primes are said to be the building blocks of all other numbers.

So on the building theme let us imagine that the RH is like a house built of bricks. We have discovered a completely built house made of bricks but we cannot work out its exact construction yet (ie solve the problem).

All we know is that the house seems to be constructed in a certain straight forward way but we're not sure whether the whole house is made using this same method or not as the house is so enormous as the defy a physical exploration of it.

The house is the RH, and the method of construction is the zeta function and critical line. As I've said, the 'house' which equates, mathematically speaking, to the sum total of all possible primes, is so enormous that we may never be able to define its size finitely. The size of the group of all possible prime numbers is debatably infinite so using super computers to check all primes may not work no matter how fast our calculations become.

So we've got a possibly infinitely sized house with a possibly infinitely adhered to construction method (critical line) so what do we do? Well, it just so happens that when we look at the house we can see that it is made up of many many bricks (primes) which regardless of size or amount all follow the same pattern of formation (i.e. all are bricks). All bricks are identical in one important aspect, they are all prime. This gives us an important inroad to studying the house (RH) as we can disregard the relative "size" of the house (which is almost impossible to comprehend accurately) and its construction method as we now know that it is built of infinitely repeating identical pieces (prime numbers).

So now we've removed the basic stumbling block to our solution. We already knew we had an extremely complex and possibly infinite idea to solve but now we know that this idea (RH) can be split up into repeating pieces that always follow the same pattern, otherwise they wouldn't be called "prime" would they, if they were not evenly divisible by themselves and 1. So now we've got the idea that understanding primes as separate from RH may help us with solving the problem.

In my answer to this problem in this thread I've explained the origin of numbers and if you read my original answer to this problem you'll see that Pro theory takes a completely different approach to most other theories as it attacks the notion of 'singularity' at its core. By this I mean that even if somebody provided a solution to the Riemann Hypothesis the opposite and neutral potentials to their solution would still have to be accounted for to be totally accurate according to Pro theory.

Solve one prime and solve them all.

We just need to explain the notion of a single "prime" number to explain them all, they are all identical in their (prime) nature of construction, only the size differs and the idea of 'size' is relative at best. The RH is just exhibiting neutrality simply speaking, the critical line with real part 1/2 is in a neutral position when seen visually.

I honestly see the Riemann hypothesis as simple and dare I say it, easy to understand in principle. It's just a mathematical problem that hasn't yet been "proven" to have a singular answer.

Pro theory suggests three simultaneous potentials at all possible moments. As I've said before any singular solution would still be subject to opposite and neutral potentials in turn ad infinitum.

The critical line is exactly that in visual terms, a line. Just a line. All we need to know is if this line continues unchangingly. Looking at this request for a singular answer critically we see that the three potentials still occur and so we completely and totally undermine the concept of "singularity" in the first place.

[Pro]

I'm not for a moment saying that singularity doesn't exist, I'm just trying to show my views on things here. The critical line is neutral as it occurs between the two axes when we look at a graph like representation of the RH.

My suggestion is that as the Riemann 'zeros' have real part 1/2 (as I understand it anyway) that 1/2 is the neutral point between zero and one.

Therefore real part potential 1/2 will exhibit neutral distributional characteristics when studied in forward sequence such as the RH. Come to think of it we could look at the Riemann zeta function in reverse or look at negative primes as well.

All things (everything) work in both forwards and backwards directions and the point at which they are not definably either potential we get zero. Incidentally, zero as a symbol is a loop, could be coincidence but it makes for an interesting aside I think.

My reason for mentioning the loop is Pro theory and its looping properties, the zero(s) are the key to understanding this problem as after all it's the zeros that seem to lie on the critical line subject to a contrary (opposite) instance.

I think that it is simple to understand the RH. All it says is that all of certain zeros lie on the critical line with real part 1/2. It is a singular statement isn't it and according to Pro theory we know that in theory there are likely to also be opposite and neutral potentials within all singular statements such as the one made by Riemann.

I can't help but see it as simple like this. It's real part 1/2 which is not real or non real it is half, neutrality in example. I think it is because zero is a neutral potential in the first place, itself symbolising the point between numbers 1 or not 1, that zeros will appear to be distributed neutrally when studied in forward sequence like Riemann chose to do.

Numbers have three possible potentials of formation I think and the key to "solving" the Riemann Hypothesis is simply to realise that zero is neutral and so is the critical line. It couldn't really be any more simple than that I don't think.

Riemann made a conjecture about an equation being either true or false and wasn't aware that in literal theoretical potential three simultaneous answers are equally possible

[Pro]

So there's a few things I could say now about numbers and the RH which after all is simply the observation of a pattern and a request for an unchanging "proof" of this pattern (ie the critical line). If we look at what makes a number "prime" we see that it can only be evenly divided by itself and 1. There should be an opposite to the prime numbers as well according to Pro theory. So we now have prime numbers, an opposite to prime numbers, and neutral (between primes in this context).

Primes are simply repeating combinations of the same three fundamental amounts (1, not 1, and zero) as is the same with all other numbers in my view. Numbers are added together, multiplied etc but in reality they still contain only repeats of the same potentials, no matter how relatively "large" or "infinite" they may seem to be.

The RH is often cited as being able to provide an explanation of all numbers isn't it. I think that if we understand numbers in the way that I stated above we can solve and understand the RH. Solving the RH involves

accepting or realising that zero is neutral, the critical line is therefore neutral, and that there are no unchanging singularities ultimately. By this I mean the fundamental idea of Pro theory and its opposite and neutral suggestions.

So the RH predicts and manifests neutrality, as zero is neutral.

If you wanted to say "is zero always neutral?" I'd also suggest three potentials but having said that it's reasonable to assume that in this specific context of the RH exposing neutrality within numbers that neutrality will always be present in the form of the critical (neutral) line. This is about as accurate as we can possibly get with a "proof" of the RH. The RH IS the critical line in effect, the absolute crux of Riemann's original statement was the critical line continuing forever.

If indeed we assume that these numbers are transcendental then we can gain an inroad to studying the problem.

Again we can see that for every even number there is an opposite. In this case the opposite would be an odd number. Let's keep things simple and just say that we have even and odd numbers. Pro theory adds neutral to the mix to complete the loop of three potentials simultaneously and then we have odd, even and neutral (zero) to deal with.

To be honest it doesn't surprise me that people associate the RH with QM and DNA sequencing as my whole point with

Pro theory is that everything shares a common pattern of structure and formation. If you look at DNA for example you see that there are two interwoven strands and nothing in the middle. There are not 10 or 50 million strands there are simply two opposite strands with neutrality in the middle. If we look deep into Quantum Mechanics we see that all atomic structure is based on protons, neutrons, and electrons. The same three potentials again.

If we sink into sub-atomic level we notice that sub-atomic particles are fractional versions of the same original charge. In other words no matter how small we go with particle physics we still see only the three fundamental potentials manifest. Anti-matter for example is the opposite of matter, the point between the two would be not quite matter or anti-matter wouldn't it. All anti-particles are opposites of actual particles.

The RH asks us "Is this conjecture unchangingly true?"

To which I provide three possibilities equally matched in potential.

The SDC asks us "Do these equations have a solution?"

To which I provide the same argument.

L and zeta functions are simply filters through which we pass values in the same way as every other mathematical function, at least this is how I see them.

Perhaps my view is too simple to be properly translated into mathematics as I use a visual method exclusively for my study of RH and I don't really refer to larger amounts than 1 (singular) More than 1 (not 1) and zero (neutral). By saying this I only mean that it doesn't seem to be worth my time to study other higher amounts of create my own commutative groups as in my view this would seem to complicate what is essentially a very simple problem (RH). Although I say this I don't mean that I'm not interested in the work of others though, I just have my ideas and stick to them personally.

I often think about the "point" at which all number problems converge If there is to be such a point anywhere within the primes it will be the neutral potential of the prime zeros used as a base from which to study RH et al. Zeros are neutral and the critical line is neutral in relation to the two axes when plotted on a simple graph and this translates into computer generated pictures of RH patterns in action.

Zeros will always be neutral won't they, pretty much anyway, so if zeros are (to quote Riemann) "very probably" neutral forever and the RH zeros lie "very probably" on the critical line we've got a proof or as near as we can get in my opinion. We have to take the looping and infinite nature of all numbers into account here obviously but this is still the correct answer or at least I think it's the most "correct" answer or "point" in the universally unchanging sense of what we know as "proof."

So to sum up I'm offering my own version of a "proof" of RH, namely that all non-trivial zeros lie continuously on the critical line ad infinitum literally because zero is the point between a number and not a number. I think this makes sense.

[Pro]

I'm editing this text as I go but I thought I'd post the majority of what I've said before so that eventually it will be able to be turned into a coherent document

[Pro]

All I'm trying to say is:

1. All numbers are based on repeating combinations of three amounts or potentials.

2. All number problems are in turn based around these three repeating potentials.

3. If we can understand and accept the three pronged nature of all things we can see that zero is neutral and so is real part one half.

4. The critical line of Riemann zeros is essentially a neutral manifestation as it's composed of neutral amounts (zeros).

5. This is why when viewed visually the RH zeros seem to lie on a line between axes, at the point in-between real and non real, i.e. real part one half.










Photo Source Article

[Pro]

I've been thinking a lot about Riemann and the primes etc lately, trying to come to some sort of semi-coherent conclusions. I find it difficult to write clearly without constant reference to opposites and the like which makes for less than easy reading at times I'm sure.

I've said before that if we can solve and/or understand a single prime number we can understand them all and by default we can also solve the Riemann hypothesis too. It's a lot more simple than currently allowed for. Let me explain.

The point of RH is to find out in terms of both abstract theory (patterns etc) and actual calculations whether or not the prime numbers have a definite and unchanging structure or not. It really is as simple as that. RH asks us a question and expects a singular answer as at the time it was first postulated (around 1859) singularities were taken for granted.

So why are prime numbers so important? And why is it often said that solving the Riemann hypothesis will also fundamentally change physics? The answer is again a simple one. Physics and mathematics are siblings, some might even say of the Siamese type. You cannot study one discipline without an inevitable cross over between the two. It works both ways, mathematics compliments physics and physics compliments mathematics.

So now we know that physics and maths are interchangeable disciplines with at least a fair amount of overlap. The easiest way to look at this relationship is probably to think of using mathematics to study and quantify physics and using the patterns and movements of physics (atomic structure et al) to visualise flows and changing patterns within mathematics.

I think it's worth noting that deciding which of the two disciplines came first in history is a bit like the Chicken And The Egg Paradox, singularly unsolvable. Humankind has always wondered and measured and calculated since time immemorial and no doubt will continue to do so.

Back to RH and the primes then and we're left wondering just why the primes are considered more important than other numbers. The answer is that primes are said to be the building blocks of all other numbers. Take this statement as literally as you like, the more so the better, prime numbers are the key to understanding all other numbers in current mathematical thinking.

This comes back to my house analogy above, if primes are the bricks of which our house is built, and natural numbers are the cement, all we have to do is understand a single brick as the house is made of many repeated bricks. Prime numbers create all other numbers because they repeat their same pattern over and over and over again. It is from this process of studying the primes in the positive (forward) sense that we derive all larger combinations. This is true regardless of the relative size or dimension of whatever number we happen to be studying.

So if we accept that primes are repeated throughout the structure of all numbers no matter how large or small, no matter whether negative or positive numbers, we can finally see why we can understand a prime of any size because it always has the prime characteristic no matter what.

It shouldn't necessarily take a load of complicated equations to come to this conclusion, all it takes is an understanding of potential as defined or at least suggested by Pro theory. Simply put this means that we need to look closely at what makes a number prime, and also what characteristics (if any) all numbers share between themselves.

A starting point for this could be a statement as simple as saying that all numbers share the same name of 'number.' Or all numbers share the same property of allowing themselves to be manipulated in certain ways, added to others for example.

Continued below...