Is it possible to simplify the RH problem?

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In summary, you are not a mathematician, and are asking for help understanding the Riemann Hypothesis. You think that zeta might represent a way to approach the problem in more heuristic terms, but you are not sure if you are right.
  • #1
PeterJ
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Hello everybody. It's my first post and I'm not a mathematician so please bear with me. I'll try to make it vaguely interesting.

I'm fascinated by the problem of deciding the Riemann Hypothesis. The trouble is, I'm not clever enough to understand it. The zeta function may as well be martian hieroglyphics and I have no idea what a diagonal lemma is.

I do have a good heuristic understanding the primes, however, and wondered whether there might be a way to understand the RH problem in more heuristic terms. This would require a massive simplification and maybe it can't be done, but at the moment I can't see why not. Complex (for me) mathematics often represents simple mechanical processes.

By 'heuristic understanding' here I mean that I know how the primes work. For a musician their behaviour is not hard to understand once the mechanism that generates them is understood. A correct (albeit not rigorous) heuristic proof of the TP conjecture is possible armed only with an understanding of multiplication. It is not the primes that are the problem for me it's the mathematics, it's translating the mechanics of the number line into equations, virtual landscapes and so forth.

I wondered whether it would be possible for me to approach the problem by reducing the zeta function to a black box. When we input a pair of numbers they are transduced into a new pair by some (for me) forever incomprehensible process. This would be a strictly deterministic process such that in principle it would be possible to backwards engineer the zeta function from a study of the behaviour of the inputs and outputs. Am I okay to think of the process in this way?

If this does actually represent the situation then the first thing I'd like to ask is what the inputs to the black box that produce the relevant zeros actually are, and which numbers have to be inputted in order to produce R's landscape. A very naive question, I know. Even asking sensible questions about the RH is difficult for a layman.

Also, would I be right to say that the zeta function acts like a resonator? Marcus du Sautoy's remarks about tuning forks and quantum resonators got me thinking. As an ex sound engineer I'm struck by the similarity between the way primes are produced and the way a plate reverb works. I even wonder whether a plate reverb might be a simple model of a quantum drum, but that's another story.

Anything that anyone can tell me about this problem that I can understand will be gratefully received. If I were younger I'd get some maths lessons but it's too late. Thanks.
 
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  • #2
PeterJ said:
I'm fascinated by the problem of deciding the Riemann Hypothesis. The trouble is, I'm not clever enough to understand it. The zeta function may as well be martian hieroglyphics and I have no idea what a diagonal lemma is.

You're pretty far away, then. Goedel diagonalization is standard undergraduate stuff, understandable by a motivated high-school student. The RH is one of the deepest problems in analytic number theory (or, indeed, of all mathematics).

PeterJ said:
I do have a good heuristic understanding the primes, however, and wondered whether there might be a way to understand the RH problem in more heuristic terms.

Sure, that's easy. The RH is equivalent to the statement that
[tex]|\pi(x) - \operatorname{li}(x)| < \frac{1}{8\pi} \sqrt{x} \, \log(x)[/tex]
for all x >= 2657. Using Cramér's heuristic model of the primes, this is true with probability 1. So heuristically, the RH true.

The hard part is bridging the gap with a proof rather than a heuristic.

PeterJ said:
I wondered whether it would be possible for me to approach the problem by reducing the zeta function to a black box. When we input a pair of numbers they are transduced into a new pair by some (for me) forever incomprehensible process. This would be a strictly deterministic process such that in principle it would be possible to backwards engineer the zeta function from a study of the behaviour of the inputs and outputs. Am I okay to think of the process in this way?

No. If we want to prove that zeta has certain properties, we can't treat it like a black box.
 
  • #3
Thanks. I realize Goedel diagonalization is standard stuff. Couldn't understand the equations, which are also probably standard stuff.

The last point seems slightly off-track since I don't want to prove that zeta has certain properties. My thought was that zeta merely reveals properties that are already encoded in the input numbers. Probably nonsense.

Am I wrong to think zeta could be recreated from an analysis of its inputs and outputs?
 
  • #4
PeterJ said:
Thanks. I realize Goedel diagonalization is standard stuff. Couldn't understand the equations, which are also probably standard stuff.

I'm sure you could understand diagonalization if you looked into it. To understand the zeta function you must minimally understand analytic continuation, since the 'defining series'... isn't. (For the regions you care about for the RH, the standard series diverges.)

PeterJ said:
My thought was that zeta merely reveals properties that are already encoded in the input numbers. Probably nonsense.

Probably. The question is just "is there a z with Re(z) > 0 such that zeta(z) = 0", which looks at all points z, not just those with specially-coded information.

PeterJ said:
Am I wrong to think zeta could be recreated from an analysis of its inputs and outputs?

I'm not sure what you mean here. A function is just a map between inputs and outputs. You don't need to use a particular symbolic form of the zeta function, if that's what you mean. On the other hand, it wouldn't be enough to look at individual points (say, using a computer to generate the value at those points) unless one was itself a counterexample; you'll need to understand how the function works in order to prove things about it.
 
  • #5
I like to think I could understand a lot of the maths, yes, given time, but I know I could never understand all that would be required for this problem. I'm in complete awe of anyone who can understand it.

I suppose I was asking if the zeta function is a map between inputs and outputs, such that each unique input will produce just one unique output and always the same one. It would follow, would it not, that the characteristics of the outputs are encoded in the inputs.

Another way of coming at it would be to ask whether we can predict which inputs will produce relevant zeros. Now I come to think of it that's what I should asked in the first place. But even this simple question may be daft. If even this question is daft I'll go away and have rethink.

Thanks for your help.
 
  • #6
PeterJ said:
I suppose I was asking if the zeta function is a map between inputs and outputs, such that each unique input will produce just one unique output and always the same one. It would follow, would it not, that the characteristics of the outputs are encoded in the inputs.

I suppose I don't know what you mean by "the characteristics of the outputs are encoded in the inputs".

PeterJ said:
Another way of coming at it would be to ask whether we can predict which inputs will produce relevant zeros. Now I come to think of it that's what I should asked in the first place.

Right, that's the whole issue. It's like saying, "the first step toward solving RH is solving RH". Yes, true -- but not very enlightening. :shy:
 
  • #7
usually mathematicians start by making things more complex before trying to find a solution to a problem. you are among the few who is trying to take the opposite path.
 
  • #8
Yes. Simplifying problems is a hobby. It works for the TPC, Russell's paradox and many other problems, (and it kept my business alive through many a crisis). I was wondering if it would work for RH. Seems highly unlikely at this point.

CRG - For you the point about inputs and outputs may not be enlightening, but I've just learned something very important from your reply.

What I meant by saying the characteristics of the outputs are encoded in the inputs is this. From a glance at a series of primes we may see little indication of pattern or rule-governed behaviour, especially if they are non-sequential. If we feed them into a function which simply squares them, however, the fact that the results always fall at 6n+1 reveals unmissable characteristics of the series that were not previously obvious. The behaviour of the outputs is encoded in the inputs and revealed by the function. Clumsy way of putting it, no doubt, but that's all I meant.

But you didn't actually say whether we can predict the relevant zeros from the inputs, either in practice or in principle. Are you saying that there's a sense in which making this prediction is the whole problem?
 
  • #9
"A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that

\sigma(n) \le H_n + \ln(H_n)e^{H_n}

for every natural number n, where Hn is the nth harmonic number, (Lagarias 2002)."

found in:

http://en.wikipedia.org/wiki/Divisor_function

I am not sure if it is a simplification or not but you can always try to prove the above result since it's equivalent to proving RH.
 
  • #10
PeterJ said:
What I meant by saying the characteristics of the outputs are encoded in the inputs is this. From a glance at a series of primes we may see little indication of pattern or rule-governed behaviour, especially if they are non-sequential. If we feed them into a function which simply squares them, however, the fact that the results always fall at 6n+1 reveals unmissable characteristics of the series that were not previously obvious. The behaviour of the outputs is encoded in the inputs and revealed by the function. Clumsy way of putting it, no doubt, but that's all I meant.

I understand what you say above, but not how it applies to the situation at hand. In your example you start from a 'mysterious' sequence (the primes), apply a function, and get a result; studying the result tells you something about the sequence. But you're suggesting, as far as I can tell, taking some really big, well-understood set (the complex numbers C, or the non-real complex numbers C \ R, or something like that), applying the zeta function, and looking at what comes out.


PeterJ said:
But you didn't actually say whether we can predict the relevant zeros from the inputs, either in practice or in principle. Are you saying that there's a sense in which making this prediction is the whole problem?

Some notation: For a set S and a function f on that set, let f(S) (the direct image) be {f(s): s in S} and let [tex]f^{-1}(S)=\{x: f(s) = x, s\in S\}[/tex] (the indirect image).

The whole problem is determining whether [tex]\zeta^{-1}(\{c\in\mathbb{C}: \Re(c)\neq1/2,\Im(c)\neq0\})[/tex] is empty or not, so in that sense yes -- if you can predict where the zero are, you're done. (Of course predicting some is not enough, you'd need to be able to predict all.)
 
  • #11
epsi00 said:
I am not sure if it is a simplification or not but you can always try to prove the above result since it's equivalent to proving RH.

I find the Lagarias problem to be a more difficult version of the Schoenfeld problem (also equivalent to the RH; I mentioned it in my first post here). But you're welcome to take a crack at it!
 
  • #12
CRGreathouse said:
I understand what you say above, but not how it applies to the situation at hand... The whole problem is determining whether [tex]\zeta^{-1}(\{c\in\mathbb{C}: \Re(c)\neq1/2,\Im(c)\neq0\})[/tex] is empty or not, so in that sense yes -- if you can predict where the zero are, you're done. (Of course predicting some is not enough, you'd need to be able to predict all.)

Thanks - even if it's all hieroglyphics to me. I realize it's a struggle to talk about this with a mathematical duffer. I was wondering whether proving the zeros behave in a certain way is equivalent to proving that the relevant inputs have certain properties. But even if this question is sensible I seem to be too far out of my depths to understand the answer. I'm not looking for a solution, of course, just exploring whether there's a more accessible route into the problem.

On a more general and chatty note. Do you believe that books such as those by Derbyshire and Du Sautoy are good non-expert introductions to number theory? I believe they are awful (albeit that they are brilliant in many ways), and wonder why nobody has written a better one. There's definitely a market for a primer but I've never come across one. What I mean by a primer is something that explains the behaviour of the primes and thus makes sense of the equations used to model it. This is what seems to be missing from every book and article that I've read, and yet it seems to be the only sensible starting point for an explanation aimed at the general reader. When people ask me to recommend a book I can't. I'm wondering why nobody is cashing in on what could be a nice little earner, and whether it's because mathematicians forget what it was like not to be one. All experts have that problem, of course, but it seems a particular problem in this context.
 
  • #13
PeterJ said:
Thanks - even if it's all hieroglyphics to me.

Sorry, I was trying to be clear. Let me try the same thing without symbols: the whole question is whether there are zeros zeta(x + iy) not on either of the lines y = 0 and x = 1/2. If we knew where all the zeros were, we'd just test to see if any were not on these lines. So knowing where all the zeros solves the problem.

Also, you can't really get anything from looking at the values that the zeta function takes on, since by Picard's theorem (read: "trust me") it takes on all complex values except possible one value.

PeterJ said:
I'm not looking for a solution, of course, just exploring whether there's a more accessible route into the problem.

Many routes are known. But the field is not yet well-developed enough that we can say which are more accessible! (There are other unsolved problems where there is a reasonably well-understood path to solving the problem, even though it hasn't been followed yet; perhaps Goldbach's weak conjecture is an example.) So that part isn't hard just for you but for everyone.

PeterJ said:
On a more general and chatty note. Do you believe that books such as those by Derbyshire and Du Sautoy are good non-expert introductions to number theory? I believe they are awful (albeit that they are brilliant in many ways), and wonder why nobody has written a better one. There's definitely a market for a primer but I've never come across one.

I haven't read their books so I don't have an opinion on that point. But I would suggest that it's hard to write a widely-accessible primer for the subject because the subject is very difficult, and writing an overview that can be understood by an 'ordinary' (smart but untrained in mathematics) person is extremely challenging.

Making math understandable is not simple by any means!
 
  • #14
Okay CRG, I've decided to book some tuition in order to get to grips with the issues and will stop bothering you. I need to take a few steps back before trying to go forward again. Many thanks for your patience. Much appreciated.

Regards
Pete
 
  • #15
Sounds good. Post again when you have new insights or questions.

I could use more complex analysis, myself...
 
  • #16
there are several operator whose eigenvalues are precisely the imaginary part of the Zeros

the main and biggest problem ,is to show how there are no zeros away the critical strip Re=1/2
 

1. Can the Riemann Hypothesis be proven?

As of now, the Riemann Hypothesis remains an unsolved problem in mathematics. Several attempts have been made to prove or disprove it, but no conclusive solution has been found.

2. How can the RH problem be simplified?

The RH problem can be simplified by finding a more general statement that implies the Riemann Hypothesis. This can help in narrowing down the focus and finding a solution to the problem.

3. Why is the RH problem important?

The Riemann Hypothesis has far-reaching implications in various areas of mathematics, including number theory, complex analysis, and prime number distribution. Its proof or disproof can lead to significant advancements in these fields.

4. What are some approaches to solving the RH problem?

There are several approaches to solving the Riemann Hypothesis, including using analytic methods, algebraic methods, and computational methods. Each approach has its own set of challenges and limitations.

5. How close are we to solving the RH problem?

As of now, there is no definitive answer to this question. While there have been some promising results and breakthroughs in recent years, the Riemann Hypothesis remains unsolved. It is a complex problem that requires further research and collaboration among mathematicians to find a solution.

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