
Mathematician-M.D. claims to have solved the 110-year-old Lindelöf hypothesis - izenme
https://viterbischool.usc.edu/news/2018/06/mathematician-m-d-solves-one-of-the-greatest-open-problems-in-the-history-of-mathematics/
======
impendia
Speaking as an analytic number theorist, the branch of math of which the
Lindelof Hypothesis is part:

This is a _huge_ deal, _if_ true. But USC's PR machine seems to have jumped
the gun.

The paper in question, found here

[https://arxiv.org/pdf/1708.06607.pdf](https://arxiv.org/pdf/1708.06607.pdf)

has so far only been posted to the arXiv (and only eight days ago). It has
presumably not been subjected to any sort of peer review yet. No third party
other than USC has announced the results. There's no chatter among my
mathematician friends, or on the blogosphere.

Fokas's results could be correct. If the community comes to a consensus that
they are, this would be a tremendous advance, and the analytic number theory
community as a whole will be trumpeting them.

But, for the time being, I stipulate that some small technical error is
probably lurking in the details, which would take hours to find, and which
will tank the proof.

I hope that I am proven wrong. Until then I propose the headline:
"Mathematician-M.D. claims to have solved one of the greatest open problems".

~~~
logicallee
[EDIT: edited this comment substantially.]

Speaking as an analytic number theorist, the branch of math of which the
Lindelof Hypothesis is part, could you tell us what you personally think of
the paper?

(Let's try to avoid just gratuitous negativity:
[https://blog.ycombinator.com/new-hacker-news-
guideline/](https://blog.ycombinator.com/new-hacker-news-guideline/) )

I read the paper after you linked it and it seems serious. Fokas is the guy
who invented the Fokas Method -
[https://en.wikipedia.org/wiki/Fokas_method](https://en.wikipedia.org/wiki/Fokas_method)

Structurally this is what my impression is of the paper (I don't understand
the math):

The result seemed very well-presented, includes background and a 2-page
inroduction, summarizes the derivation of the main result from page 5-12,
derives its main theorems and lemmas used, and then from p. 50-52 summarizes
it all again. Finally three appendices in 4 pages provide some numerical
verification (a sanity check) and an acknowledgment section says "This project
would not have been completed without the crucial contribution of Kostis
Kalimeris. Kostis has studied extensively the classical techniques for the
estimation of single and multiple exponential sums; these techniques are used
extensively in our joint paper with Kostis [KF] and some of the results of
this paper are used in section 6. Furthermore, Kostis has checked the entire
manuscript and has made important contributions to the completion of some of
the results presented here." There are another 7 paragraphs of acknowledgments
going back more than 3 years, and finally 3 pages of references, including to
private correspondence and preprints.

The affiliations on the paper are Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, and Viterbi School of
Engineering, University of Southern California.

Additionally, this researcher has a proven track record, his Wikipedia article
says:

>He has made seminal contributions in a remarkably broad range of areas which
include: symmetries, integrable nonlinear PDEs, Painleve' equations and random
matrices, models for leukemia and protein folding, electro-magneto-
enchephalography, nuclear imaging, and relativistic gravity. Also, he has
introduced a completely new method for solving boundary value problems known
as the Fokas method, which has been acclaimed as the most important
development in the analytical treatment of PDEs since the introduction of the
Fourier transform

He is the winner of the Naylor prize (past winners include Roger Penrose, 1991
and Stephen Hawking, 1999, among others) and holds 7 honorary doctorates
(right side of his Wikipedia page.)

~~~
yxhuvud
No, it is the proper response to unverified claims of scientific breakthrough.

------
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..

~~~
fifnir
Isn't MD for..like.. medical doctors ?

If he's an MD but without a medical degree (which I guess is the case), then
what's the difference betweeen MD, PHD or 'professor' ??

~~~
naturalgradient
So this goes off on a tangent but I feel it relates to noncentrality [0].
Fokas has a PhD in maths. Being an MD or having gotten an MD 40 years ago is
clearly entirely non-central to his career. Calling him Mathematician-MD seems
like it is meant to make him seem a lesser mathematician, e.g. by insinuating
that this is just something he does part time, and that he can hence be taken
less seriously.

I don't know what the poster meant by suggesting 'Mathematician-MD', but it
reads weirdly to me for that reason. It's highlighting an attribute of a
person that is entirely unrelated to his career or this article. Why if not to
denigrate him? The title should be changed to neutrally reflect his position.

[https://www.lesswrong.com/posts/yCWPkLi8wJvewPbEp/the-
noncen...](https://www.lesswrong.com/posts/yCWPkLi8wJvewPbEp/the-noncentral-
fallacy-the-worst-argument-in-the-world)

~~~
hypeibole
I actually believe the intent is the opposite.

I think the PR people are just trying to sell Fokas as a polymath genius.
"Look, he is not only a mathematician but also an MD, wow!"

~~~
naturalgradient
Thank you, I had not considered that. I think researchers or a HN audience may
perceive this differently than the average reader.

~~~
TomVDB
FWIW: my initial instinctive reaction to the MD part was most definitely
negative.

------
sometimesijust
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.

~~~
filmor
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.

~~~
hansbo
It was a fascinating read. Reminded me of this XKCD:
[https://xkcd.com/386/](https://xkcd.com/386/)

------
cbames89
Can someone eli5 the lindelof hypothesis?

~~~
impendia
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
arithmetic a certain way, you get -1/12.

The original definition I gave is valid when s is a complex number with real
part greater than 1. But the Riemann zeta function can be proved to have
_analytic continuation_ : zeta(s) makes sense for any complex number s, other
than 1. For example, zeta(-1) really equals -1/12.

The zeta function is easy to understand when the real part is greater than 1:
the formula I described is enough. Because of the so-called _functional
equation_ , it is also easy to understand when the real part is less than 0.
But it is in the middle that all of its secrets lie. For example, the
notoriously unsolved _Riemann Hypothesis_ stipulates that the "nontrivial"
zeroes all have real part 1/2.

The Lindelof Hypothesis stipulates that the zeta function grows very slowly
along this line (real part = 1/2). It is very closely related to the Riemann
Hypothesis. More technical, and of less direct interest to nonspecialists, but
in the same family of problems.

As an example of how much mathematicians care about this, here are the Google
search results for "subconvexity bound":

[https://www.google.com/search?q=subconvexity+bound](https://www.google.com/search?q=subconvexity+bound)

A "subconvexity bound" is any result which approaches the Lindelof Hypothesis,
for either the Riemann zeta function or a more general "L-function". A lot of
ink has been spilled on proving results weaker than what Fokas is claiming.

~~~
taneq
> 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
it", this doesn't mean the simple answer is wrong, it means you did something
else wrong (like the hidden divide-by-zero present in your typical "proof that
1 = 2"). Paradoxes point to an error in the formulation of the question.

~~~
rfurmani
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 a definition that is compatible with
most of what one would want from a limit. For example, taking running
averages, or taking it as the coefficients of a series and taking a limit. 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\. Or if
you use the fact that 1+x+x^2+x^3+... = 1/(1-x) and take x=-1 you would get
1/2\. Or a myriad other approaches that are perfectly valid with standard
convergence, and just all happen to get you that 1-1+1-1+... is 1/2\. And yes
if you squint very hard you would get that 1+2+4+8+16+... = 1/(1-2)=-1

~~~
taneq
> 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?

~~~
Doxin
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

~~~
Koshkin
> _In this case it trends solidly in the direction of 1 /2_

...except that it doesn't; to me, it even looks more like it trends in the
opposite direction, i.e. it is trying to stay away from 1/2.

~~~
whatshisface
The farthest it could get from 1/2 would be if it ran off to + or - infinity.
Hovering between +1 and -1 could be compared to an orbit.

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

~~~
robertelder
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 of cryptography faster, or
apply it to building newer and stronger cryptography.

~~~
hsienmaneja
So, people hire you to break into their places... to make sure no one can
break into their places?

~~~
zinckiwi
A Sneakers quote. Worth watching for Hollywood's take on the early 90s version
of our current social media/social engineering quagmire.

------
jvln
At the acknowledgement the only big name from the field is Peter Sarnak
[https://en.wikipedia.org/wiki/Peter_Sarnak](https://en.wikipedia.org/wiki/Peter_Sarnak).
If Peter Sarnak vouch for the result it might be correct.

------
danharaj
I'm skeptical

