If you want to understand it - well, it can be understood at many levels, can get a rough idea of what it is about right away.
See Michael Ross's answer here.
Michael M. Ross' answer
This answer is about - what if you want to have a go at thoroughly understanding it - and proving it - using all the techniques of modern maths.
So - you might well end up studying a large part of modern pure and applied mathematics with that as your goal.
Absolutely required would be:
That much is needed to properly understand the hypothesis.
For a bit of context - I'm a mathematician by training - and did study those topic areas also - but specialised in maths foundation and logic - and I'd say that I don't really understand the Reimann hypothesis :).
We didn't cover it in detail - I did a one year course in number theory but that was along with four other topics - set theory and logic (single course), algebraic topology, fourier analysis, and group theory - that was in the final year of my maths degree 40 years ago. I still remember them well as they were exciting and interesting topics for me.
So you can imagine it wasn't that thorough.
Actually I studied exactly the topics needed to go on and do research into the Reimann hypothesis, come to think of it + 2 others not absolutely needed so much but would be a good idea for anyone in pure maths to do those anyway, set theory and logic, and group theory. But I didn't study them in enough depth.
So, I know what an analytic continuation of a complex function is (technique that lets you prove that a function you have that is defined only over a particular domain can be "continued" in a continuous differentiable way over a much larger range by piecing lots of bits of functions together - and that the continuation is always unique - elegant result in complex analysis - but often not easy to construct).
Also, of course what zeros are and what the prime number theorem is. I can understand the statement of the theorem. But wouldn't say I properly understand the context and implications which is what makes it so interesting to mathematicians - so in that sense I don't understand it,
You would need to have a firm basis in all this stuff and know it inside out to have a chance at it :). Otherwise you don't really know what it is you are trying to prove, in a way.
But that's just a start. If you look at a survey of the attempted solutions here:
Riemann hypothesis
Then they mention Hamiltonians (area of applied maths used in physics), statistical mechanics, non commutative geometry, Hilbert spaces, quasicrystals, and elliptic curves, just to list a few that catch the eye.
Of course you don't need to know them all, who knows, maybe you just learn the theory of quasicrystals and come up with some cool new idea that applies them to the Reimann Hypothesis, and you are done...
So - hard to give any advice.
Go to university, do a good undergraduate degree, make sure you cover topology, number theory, and complex analysis. Tell your tutors about your interest and they may suggest other courses to follow.
And - at the end of it - expect to add your name to the list of unconfirmed proofs of the Reimann hypothesis Search for Reimann Hypothesis proofs
And eventually - chances are someone finds a flaw in your first proof and you have to start again.
Statistically your chances of success don't seem good - with thousands of brilliant mathematicians attempting it and failing - but you know that already.
You'll learn a huge amount about maths though in the process. So if you are not that concerned about succeeding at the end, can be a good way to motivate yourself to learn lots of maths.
Not at all want to discourage you. Great to have dreams and aims. Especially if you are not too discouraged if they take longer than expected or are not achieved at all. If the main thing is the journey especially - rather than achievement and success at the end - then can be great.
And - as you say - is a remote chance - that maybe like Andrew Wiles with Fermat's last theorem, you eventually do find a proof. Wiles' proof of Fermat's Last Theorem. It took seven years of his life, full time, and when he explained his proof it took many lectures just to present it in its entirety, and that would be with skipping over many details that would be known to his audience or reasonably easy for them to see.
Which doesn't mean that it has no easier proof, like one of say ten pages instead of hundreds of pages. Who knows if it does or doesn't. But- for whatever reasons, if there is an easy proof, seems that to humans at least, it's elusive and hard to find.
See Michael Ross's answer here.
Michael M. Ross' answer
This answer is about - what if you want to have a go at thoroughly understanding it - and proving it - using all the techniques of modern maths.
So - you might well end up studying a large part of modern pure and applied mathematics with that as your goal.
Absolutely required would be:
- Number theory- that is to say theory about the properties of numbers
- In particular, thorough understanding of the Prime number theorem
- Topology
- Complex number - numbers that involve the square root of -1.
- Complex analysis - study of continuous differentiable smooth curves over complex numbers
- Analytic continuation - this is part of Complex analysis but highlighting it as it is absolutely required that you understand this thoroughly just to understand statement of the Reimann hypothesis
- Obviously lots of preparatory earlier stuff like geometry, calculus etc - basic maths education.
That much is needed to properly understand the hypothesis.
For a bit of context - I'm a mathematician by training - and did study those topic areas also - but specialised in maths foundation and logic - and I'd say that I don't really understand the Reimann hypothesis :).
We didn't cover it in detail - I did a one year course in number theory but that was along with four other topics - set theory and logic (single course), algebraic topology, fourier analysis, and group theory - that was in the final year of my maths degree 40 years ago. I still remember them well as they were exciting and interesting topics for me.
So you can imagine it wasn't that thorough.
Actually I studied exactly the topics needed to go on and do research into the Reimann hypothesis, come to think of it + 2 others not absolutely needed so much but would be a good idea for anyone in pure maths to do those anyway, set theory and logic, and group theory. But I didn't study them in enough depth.
So, I know what an analytic continuation of a complex function is (technique that lets you prove that a function you have that is defined only over a particular domain can be "continued" in a continuous differentiable way over a much larger range by piecing lots of bits of functions together - and that the continuation is always unique - elegant result in complex analysis - but often not easy to construct).
Also, of course what zeros are and what the prime number theorem is. I can understand the statement of the theorem. But wouldn't say I properly understand the context and implications which is what makes it so interesting to mathematicians - so in that sense I don't understand it,
You would need to have a firm basis in all this stuff and know it inside out to have a chance at it :). Otherwise you don't really know what it is you are trying to prove, in a way.
But that's just a start. If you look at a survey of the attempted solutions here:
Riemann hypothesis
Then they mention Hamiltonians (area of applied maths used in physics), statistical mechanics, non commutative geometry, Hilbert spaces, quasicrystals, and elliptic curves, just to list a few that catch the eye.
Of course you don't need to know them all, who knows, maybe you just learn the theory of quasicrystals and come up with some cool new idea that applies them to the Reimann Hypothesis, and you are done...
So - hard to give any advice.
Go to university, do a good undergraduate degree, make sure you cover topology, number theory, and complex analysis. Tell your tutors about your interest and they may suggest other courses to follow.
And - at the end of it - expect to add your name to the list of unconfirmed proofs of the Reimann hypothesis Search for Reimann Hypothesis proofs
And eventually - chances are someone finds a flaw in your first proof and you have to start again.
Statistically your chances of success don't seem good - with thousands of brilliant mathematicians attempting it and failing - but you know that already.
You'll learn a huge amount about maths though in the process. So if you are not that concerned about succeeding at the end, can be a good way to motivate yourself to learn lots of maths.
Not at all want to discourage you. Great to have dreams and aims. Especially if you are not too discouraged if they take longer than expected or are not achieved at all. If the main thing is the journey especially - rather than achievement and success at the end - then can be great.
And - as you say - is a remote chance - that maybe like Andrew Wiles with Fermat's last theorem, you eventually do find a proof. Wiles' proof of Fermat's Last Theorem. It took seven years of his life, full time, and when he explained his proof it took many lectures just to present it in its entirety, and that would be with skipping over many details that would be known to his audience or reasonably easy for them to see.
Which doesn't mean that it has no easier proof, like one of say ten pages instead of hundreds of pages. Who knows if it does or doesn't. But- for whatever reasons, if there is an easy proof, seems that to humans at least, it's elusive and hard to find.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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