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Quantum physics sheds light on Riemann hypothesis (bris.ac.uk)
89 points by prateekj on Oct 13, 2013 | hide | past | web | favorite | 31 comments



Douglas Adams may have been onto something.

> Now, there are certain attributes of the Riemann zeta function called its moments which should give rise to a sequence of numbers. However, before the Seattle conference, only two of these moments were known: 1, as calculated by Hardy and Littlewood in 1918; and 2, calculated by Ingham in 1926.

> The next number in the series was suggested as 42 by Conrey (now also at Bristol) and Ghosh in 1992.

> The challenge for the quantum physicists then, was to use their quantum methods to check the number 42 and to calculate further moments in the series, while the number theorists tried to do the same using their methods.

> Prof Jon Keating and Dr Nina Snaith at Bristol describe the energy levels in quantum systems using random matrix theory. Using RMT methods they produced a formula for calculating all of the moments of the Riemann zeta function. This formula confirmed the number 42.


Sadly, he said himself that he picked this number with nothing else in mind that "a simple smallish number".

ref: http://scifi.stackexchange.com/questions/310/why-did-douglas...


Or the number picked him :D


The connection was first made (decades ago) in a chance interaction between Hugh Montgomery (who was working on the Riemann Hypothesis) and Freeman Dyson.

For a description of how serendipity struck, and an nice explanation of how scientists are trying to understand and work the analogy, see here -- http://www.americanscientist.org/issues/id.3349,y.0,no.,cont...


Really exciting connection, and I am glad I read the article. What I hate though is the box at the end of the article:

   So why is this work so important?

   As we have said, prime numbers are the basic building blocks of mathematics. 
   And primes are vital to cryptography and therefore to the ever-burgeoning world 
   of online commerce and security.
Nope, cryptography is not why this work is important. What a lot of bull.


Its a reason the average person can see value in. Since it's the average person's tax money often funding this stuff, it makes sense that academics try to explain uses that benefit the people paying for it.

While I agree that its worth it just for its knowledge pushing value alone, academics and researchers do need to take the public with them, especially these days when finances are squeezed and more people are struggling to make ends meet.


I know WHY the box says what it says. But it is nevertheless bull. If you want to improve online security, the LAST thing you should put money into is research into the Zeta function.

So, if this is what you are telling the tax payer, then you are just telling lies to the tax payer.


If there exists a fast factoring algorithm that breaks RSA, then investing in research to discover that would allow us to design and switch to better cryptosystems. If there does not exist one, discovering that fact would allow us to prove security guarantees about RSA; either way, we improve security.

If we avoid such research out of fear that we might discover something that breaks RSA, we are relying on security by obscurity.


Nobody is 'avoiding' that research. People already know a fast quantum algorithm exists for the factorization problem, which is the core of RSA. hence, people are already working on cryptosystems more computationally complex than RSA. Hell, even more computationally intensive than what quantum computers are thought to be able to do. The class of algorithms called lattice based cryptography are known to be impossible for quantum computers to break, even though these are fully classical methods of cryptography.


So where exactly should we have stopped doing math, in order to remain optimally secure for all time?


I had the same thought: One of the deepest mysteries of mathematics is important because.... of online e-commerce?

Nice to know we've got our cultural priorities straight!


There can be no higher "cultural priority" than closing the intellectual feedback loop between research and the people who pay for it. It drives our progress as a society.


Exactly. And that's why you shouldn't lie but tell the truth.


Do people really consider prime numbers the basic building blocks of mathematics? Isn't that supposed to be sets?


I think people like to say the prime numbers are the basic building blocks of arithmetic (or numbers), and we say this because of the Fundamental Theorem of Arithmetic (which says that every natural number has a unique factorization of primes).


I would have thought it would be Peano's axioms for natural numbers or whatever axiomatic system is being used; depends on what area of mathematics one is studying.


Does the riemann hypothesis even have ground breaking applications to crypto?


Not necessarily. The methods used to confirm the Hypothesis could be used in other related fields. The techniques used to solve the Hypothesis could potentially be applied to break RSA. I've read there are similarities between the Riemann Hypothesis and RSA's workings.


from what i've heard attempts to prove the Riemann hypothesis this way have been a dead end. mathematicians agree the heuristics concerning random matrices and quantum chaos are true but

Physics of the Riemann Hypothesis http://arxiv.org/abs/1101.3116

Quantum chaos, random matrix theory, and the Riemann zeta-function http://www.math.harvard.edu/~bourgade/papers/PoincareSeminar...

The quantum physics methods being used to solve the Riemann hypothesis can solve easier number theory problems as well. We can look at the structure of the primes more directly.

http://en.wikipedia.org/wiki/Apollonian_gasket#Integral_Apol...


Okay, yes, random matrix theory is a type of mathematics that was developed largely for its applications in physics. But it's still a branch of elementary mathematics! You don't need to know anything about quantum mechanics to wonder about the average eigenvalues of a random matrix.

I was hoping this was a case of a quantum physics experiment shedding light on the Riemann hypothesis -- now that would be impressive! And actually not that far fetched, either, although clearly beyond the state of the art (see: quantum computing).


From an Euler problem (412 IIRC), I had played around with Young's tableaux, counting which also gives rise to the sequence 1,2,42,24024. More numbers here: http://oeis.org/A039622

Am I onto something? Let's see if the next moment is 701149020.


I find science and math really interesting without actually knowing anything deep.

It's just like magic. There are these interesting ratio, numbers, series, and functions appear in both nature and mathematics. Similar to how scientists praise Big Band, a lot of things are so well-defined, well put together with a precise amount (in the case of Big Bang a slight off amount might actually destroy today's universe). Sometimes I have to say and assume there is this powerful being God there writing this novel...


Well, for starters we only think it's so "precise" because that's the model we've constructed to understand the laws of physics. But once we fully understand what's happening at the quantum level, we might scrap the Standard Model altogether, and discard it as useless (for whatever we plan to make next with the new found knowledge).

Plus, it may be this "precise" in the same way Earth has "exactly" the things it needed to create life, and ultimately us, humans. That's to say it wasn't precise or exact at all. It was just a potential combination of stuff, out of trillions and trillions of other combinations, which may or may not have resulted with the same things.

So who's to say our universe isn't just one of the potentially trillions of trillions of other universes out there, and that other universes exist with life-forms that we can't even imagine (inter-dimensional/inter-universal beings, etc). We just turned out the way we did because that's what we "got" at the roll of dice (sort of speak).


Well, I doubt we will scrap it altogether. There is still value in Newton despite Einstein.


I'm not reliegious, but even Thomas Aquinas disagreed at least as to math and logic, if not the universe:

"Since the principles of certain sciences, such as logic, geometry and arithmetic are taken only from the formal principles of things, on which the essence of the thing depends, it follows that God could not make things contrary to these principles. For example, that a genus was not predicable of the species, or that lines drawn from the centre to the circumference were not equal, or that a triangle did not have three angles equal to two right angles."


Cultural note: Bristol University have some nice cryptography stuff going on. They also have links to GCHQ via the Heilbronn Institute.

(http://www.bristol.ac.uk/engineering/research/research-group...)

(http://www.maths.bris.ac.uk/research/heilbronn_institute/)


Very Exciting.

I recently watched a talk by Ed Witten where he said that he thought that quantum physics would be useful for number theory eventually, it's cool to see this borne out in the present day.

Here is the talk (Knots and Quantum Theory) at the moment during the Q&A where he was asked about this. http://youtu.be/8nA17Id4JyU?t=45m3s


Suggesting that prime numbers are the atoms of arithmetic seems inappropriate. Can anyone explain this? Am I looking at a journalism major's summary of research?


I'm not sure if he originated the phrase, but Marcus du Sautoy, a professor of Mathematics at Oxford, used the phrase in his work "The Music of the Primes":

"It remains unresolved but, if true, the Riemann Hypothesis will go to the heart of what makes so much of mathematics tick: the prime numbers. These indivisible numbers are the atoms of arithmetic. Every number can be built by multiplying prime numbers together. The primes have fascinated generations of mathematicians and non-mathematicians alike, yet their properties remain deeply mysterious. Whoever proves or disproves the Riemann Hypothesis will discover the key to many of their secrets and this is why it ranks above Fermat as the theorem for whose proof mathematicians would trade their soul with Mephistopheles."


By the way, if you are at all interested in this subject, even in the slightest, I highly recommend reading Music of the Primes.


Atoms are tightly bound and hard to split, they are even harder to build. Up one level and starting with atoms the ease with which the next structures (molecules) can be created goes up tremendously (of course with a lot of variation depending on which atoms are used).

Prime numbers are can only be split (factorized) in one axiomatic way (1 and the prime itself). This is with multiplication(or it's inverse division) as the building (splitting) operation. Primes are tough to deal with, yet with them you can create (through multiplication) all other numbers. This is the fundamental theory of arithmetic.

The correlation is apparent. Though as explained with atoms we are creating structures that are made up of atoms but are not classified as atoms themselves (fusion can give you atoms from atoms but it is not easy business), but with primes numbers we make other numbers that are not primes.




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