
Physicists attempt to prove the Riemann hypothesis - seycombi
https://www.quantamagazine.org/20170404-quantum-physicists-attack-the-riemann-hypothesis/?platform=hootsuite
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jordigh
You have to be very careful with these "plausibility arguments" that
physicists like to make. It may well happen, for example, that there are
counterexamples to the Riemann hypothesis, but that they are so rare that they
have probability zero (similar to, for example, the zero probability that a
random real number in some interval is a rational number). So, a plausibility
argument saying how unlikely counterexamples are would prove nothing. Or, it
could happen, like the famous disproved Riemannian conjecture on the crossing
of pi(x) and Li(x) (see: Skewe's number) that the counterexample exists but it
is ridiculously large. The usual kind of ethos in physics is to be a bit too
loose with approximations or to disregard very extreme cases.

I don't understand the particulars of the spectrum of this operator that they
have constructed, but I have heard others describe this approach as a simple
reformulation of one hard problem in terms of another equally hard problem.

~~~
hn_throwaway_99
I'm not a mathematician, nor a physicist, but I don't understand this
statement: similar to, for example, the zero probability that a random real
number in some interval is a rational number. Is it just a consequence of
rational numbers being countable and real numbers not being countable? Seems
like an odd way to define probability in this instance.

~~~
alextheparrot
It comes from how we define infinity/the number line. Any number divided by
infinity is 0, that's easy. But between any two points there is an infinite
number of rational numbers and between any two rational numbers you can derive
an infinite number of irrational numbers. I know this isn't kosher, but it is
pretty much infinity/(infinity ^ infinity). That leads us back to the
probability of picking a rational number as 1/infinity, because for every
rational number there are infinity irrational numbers that we also could have
chosen.

David Foster Wallace had a really good book called Everything and More where
he explored the history of infinity. Might be of interest to you.

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SomeStupidPoint
That's a _really_ bad argument, since between any two irrational numbers,
there's an infinite number of rational numbers.

Ed: For an explicit construction of such numbers --

Suppose we have a<b, two irrational numbers.

Let b-a = d, the difference between them.

Then there exists N, an integer, such that 10^(-N) < d.

Let b1 be b truncated at the (N+1)th digit past the decimal point.

Then b1 is rational (finitely many digits) and b-b1 < d (and b1 < b), so b1 is
in (a,b).

You can then add back one digit of b at a time, to get b2, b3, etc which form
an infinite sequence of rational numbers converging to b from below. (We get
infinite _unique_ numbers from the fact b is irrational. Not all the b_n need
be distinct.)

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heimatau
I don't understand why Quanta magazine keeps getting promoted on HN. Guys,
it's sensational writing. Not scientific. Please understand this. Quanta is
like Wired. Poor academic quality and sensational writing. Neither are based
in scientific rigor.

I'm majoring in Pure Maths and this is annoying to see yet another poor
scientific article on Math.

~~~
thanatropism
Was that article on symplectic geometry in Quanta?

My work is with algorithms that simulate conservative systems and that sounded
pretty good to my ears. Of course they were with the human angle by the end,
but still they did a damn good job in explaining what a "cotangent bundle" is.

~~~
heimatau
My bad. Hopefully this [1] comment and this [2] one helps clear up my
position. They are accurate, yes. But poor in the sense that it's all
conjecture, not evidence based.

1 -
[https://news.ycombinator.com/item?id=14046334](https://news.ycombinator.com/item?id=14046334)
2 -
[https://news.ycombinator.com/item?id=14046131](https://news.ycombinator.com/item?id=14046131)

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jesuslop
Now and since decades ago also, see
[https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_con...](https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture)

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justifier
> As mathematicians have attacked the hypothesis from every angle, the problem
> has also migrated to physics.

eyeroll.. loose use of 'every' is the kind of overreaching probability
mathematical rigor eschews

quantum mechanics still relies heavily on probability theory and reimann has
been probably correct since its inception

it is my intended inference that a mathematical model that maps qm will be
mappable to reimann, and without any forgiveness of strict symmetries

~~~
posterboy
What's the motivation for your intention? What is a reimann? The mathematician
was called Riemann. Do you mean the Zeta function and it's variants?

~~~
justifier
i'm sorry if i was unclear.. i am aware of who bernhard reimann is and was
using his name as shorthand for the hypothesis originating in his 1859
paper(o): .."and it is very probable that all roots are real."; of which this
entire article is about

what's my motivation for my intention? I am unsure exactly.. personally? I
derive significant joy from working on mathematics; socially? to solve the
problem would solidify assumed validity across a varying subset of mathematics
and its consequents

(o)
[http://www.claymath.org/sites/default/files/ezeta.pdf](http://www.claymath.org/sites/default/files/ezeta.pdf)

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ngcc_hk
The article is very interesting and serve one purpose very well - trigger my
curisoity of what is that. Both heard of. Not understand much. But somehow it
is linked. And strangely talked in a way not totally out of reach.

If someone like me with only some basic maths/stats/QM background and interest
to know more, any pointer to understand this.

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akeck
This reminds me of story I heard about a pure math PhD student who wrote a
deep proof for their dissertation. Unfortunately, they relied on certain math
shortcuts that only work in a physics context. Their advisor didn't catch it,
and I recall it ended badly for both the student and the advisor.

~~~
heimatau
That sounds bizarre. Pure Math is more theoretical and has less limitations
than that of Theoretical Physics. Maybe this person you knew just went down
the wrong path and should've focused on Theoretical Physics and not math.

~~~
thanatropism
If I understand the parent comment correctly, this is the advanced version of
"proofs" of propositions in economics that just assume that dy/dx is defined
in such a way that dy/dz = (dy/dx)(dx/dz). A mathematician would say "Let y:
U\to V where U,V are open subsets of R^k and y\in C^1 (or C^\infty). Then..."
and define clearly what's a function, what's a variable/number/vector and so
forth.

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eastWestMath
When all you have is a golden hammer...

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emucontusionswe
These physicists observations won't achieve anything other than to corroborate
the existing hypothesis. Proof requires a lot more than just observation.

~~~
giomasce
There are no physicists' observations here, as I get it, it is all theoretical
work, which is essentially mathematics. It is not rigorous mathematics,
otherwise we would actually have a proof, but just a proposed reasoning that
in time might be made rigorous and become a real proof. Future will tell us.

