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Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

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Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#41

Post by adrianN »

throwaway0255 wrote: post #30
Cyph0n wrote: post #25

Earlier quoted context omitted.

That's just not how pure mathematics works. For instance, imagine asking that same question about Euler's totient function 50 years before RSA was developed. I am sure that there are countless other examples where a seemingly useless theorem aided in a practical problem decades or centuries later.

Now I and other non-mathematicians can look into Euler's totient function and RSA, and learn more. Thanks for answering. So is there really never a proof in pure mathematics with more obvious or immediate near-term applications? Are there any historical examples of that? A major breakthrough in mathematics that once understood, it was immediately obvious that it was going to make X work better, and then it did?

Mathematics with immediate payoff is usually called physics or computer science. For example the simplex method for linear programming made a lot of things more efficient.

Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#42

Post by hansbo »

filmor wrote: post #36
sometimesijust wrote: post #33

Hypothesis: As the complexity of proofs approaches the limits of human ability to understand, saying it is a proof becomes more important than proving it is a proof. Evidence: The Wikipedia page for the Lindelöf hypothesis already unambiguously states that it has been formally proved.

If you check the history and talk of that page you will see that there is one very persistent user who has repeatedly re-added this section while at least to others tried to remove it.

It was a fascinating read. Reminded me of this XKCD: https://xkcd.com/386/

Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#43

Post by taneq »

rfurmani wrote: post #32
taneq wrote: post #28

Earlier quoted context omitted.

> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…

But this isn't just summing up a few positive integers, in which case there's no ambiguity in what the answer is. Once you start summing up infinitely many things you have to bring in some theory and some techniques to justify what the answer is. These techniques generally have a limited scope, but there's a big theory of "divergent series" that shows that if you extend these techniques to more contexts you still get…

> So classically 1-1+1-1+1... doesn't converge, but if you "squint" and take the running averages of the partial sums (1,0,1,0,...) you would get 1/2

Why would you take the running average and how is that relevant to the sum?

Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#44

Post by Doxin »

taneq wrote: post #43
rfurmani wrote: post #32

Earlier quoted context omitted.

But this isn't just summing up a few positive integers, in which case there's no ambiguity in what the answer is. Once you start summing up infinitely many things you have to bring in some theory and some techniques to justify what the answer is. These techniques generally have a limited scope, but there's a big theory of "divergent series" that shows that if you extend these techniques to more contexts you still get…

> So classically 1-1+1-1+1... doesn't converge, but if you "squint" and take the running averages of the partial sums (1,0,1,0,...) you would get 1/2 Why would you take the running average and how is that relevant to the sum?

He's taking the running average of the partial sums:

    1-1 = 0
    0+1 = 1
    1-1 = 0
    0+1 = etc etc
The partial sum would end up being equal to the final sum by definition when you're done summing all items. Since we're talking about infinite sequences you're never done summing all items so you'll have to do something else to end up with an answer. for example seeing which way the partial sum trends. In this case it trends solidly in the direction of 1/2

Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#45

Post by Ar-Curunir »

robertelder wrote: post #5
hsienmaneja wrote: post #3

What exactly is the application for cyber security? How does this affect cryptography?

I think it's because any information we gain about the Riemann Hypothesis (Lindelöf hypothesis is implied by RH) gives us information about the distribution of prime numbers. Any time you gain information about the distribution of prime numbers you immediately gain information that can be applied to any form of cryptography that makes use of prime numbers. You could use this information either to break existing forms…

No, this is not true. Before this conjecture was proved, you could just assume it was true and see if it would lead anywhere. People already do that with the GRH. If it did have consequences, people would use the resulting algorithm regardless of whether the conjecture was true, because we believe to be almost certainly true. But even if you didn't believe that, you could see if the algorithm was effective by applying it to real world instances.

Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#47

Post by mtzet »

QML wrote: post #29
robertelder wrote: post #5

Earlier quoted context omitted.

I think it's because any information we gain about the Riemann Hypothesis (Lindelöf hypothesis is implied by RH) gives us information about the distribution of prime numbers. Any time you gain information about the distribution of prime numbers you immediately gain information that can be applied to any form of cryptography that makes use of prime numbers. You could use this information either to break existing forms…

We already have polynomial-time algorithm for primality testing [1]; not sure if a proof of the Riemann Hypothesis will affect cryptography by that much. [1] https://en.wikipedia.org/wiki/AKS_primality_test

While I agree that a proof of the Riemann hypothesis is unlikely to matter for cryptographical purposes, neither does the AKS primality test. As with many asymptotically efficient algorithms, the constants are simply too large for it to be practical.

Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#48

Post by ur-whale »

taneq wrote: post #28
impendia wrote: post #11

Earlier quoted context omitted.

The Riemann zeta function is the function zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + .... For example, zeta(2) = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = pi^2/6. As a partially tongue-in-cheek example, zeta(-1) = 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12. Obviously it doesn't make sense to add all the positive integers (the series doesn't converge ), but if you squint and ignore this, and just do the arithmeti…

> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…

I wish impendia had left out of his otherwise excellent explanation the "squinting" part and stuck to explaining that the zeta function is defined by the infinite series where that series converges, and defined where that series diverge using a tool called analytic continuation.

In short:

Where the series obviously converge, use the summing formula.

Where the series is mis-behaving, use analytic continuation instead of resorting to weird infinite series re-ordering tricks (which I've always felt to be borderline offensive from a mathematical rigor pov).

My understanding of analytic continuation is that if a function f of the complex plane is sufficiently well behaved on a certain domain of the plane, it can be "extended" to the rest of the plane in a unique way that preserves the well-behavedness.

In the case of zeta, it can be shown that zeta obeys a functional equation that allows it to be extended everywhere.

A better explanation than mine is here:

https://math.stackexchange.com/questions/437883/what-is-the-...

Re: Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis

#50

Post by naturalgradient »

Just pointing out that interestingly Cambridge, where Fokas is a professor, has not released anything.

He is merely visiting USC so it strikes me as weird that they would claim this PR so quickly.

Also Mathematician-MD somehow makes it sound like the MD means he is a lesser mathematician or not a full mathematician. Fokas is a well respected Professor at one of the top applied Maths departments in the world. A better and less biased title would be 'Math Professor' or 'Cambridge math professor' claims..

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