
The Riemann Hypothesis - gballan
https://golem.ph.utexas.edu/category/2019/09/the_riemann_hypothesis_part_1.html
======
Cybiote
Here's a quite friendly elucidation by the inimitable Avi Wigderson, from:
[https://www.ias.edu/ideas/2009/wigderson-randomness-
pseudora...](https://www.ias.edu/ideas/2009/wigderson-randomness-
pseudorandomness)

> Let’s elaborate now on the connection (explained on the cover of this issue)
> of the Riemann Hypothesis to pseudorandomness. Consider long sequences of
> the letters L, R, S, such as

> S S R S L L L L L S L R R L S R R R R R S L S L S L L . . .

> Such a sequence can be thought of as a set of instructions (L for Left, R
> for Right, S for Stay) for a person or robot walking in a straight line.
> Each time the next instruction moves it one unit of length Left or Right or
> makes it Stay. If such a sequence is chosen at random (this is sometimes
> called a random walk or a drunkard’s walk), then the moving object would
> stay relatively close to the origin with high probability: if the sequence
> was of n steps, almost surely its distance from the starting point would be
> close to √n. For the Riemann Hypothesis, the explicit sequence of
> instructions called the Möbius function is determined as follows for each
> step t. If t is divisible by any prime more than once then the instruction
> is Stay (e.g., t=18, which is divisible by 32). Otherwise, if t is divisible
> by an even number of distinct primes, then the instruction is Right, and if
> by an odd number of distinct primes, the instruction is Left (e.g., for
> t=21=3x7 it is Right, and for t=30=2x3x5 it is Left). This explicit sequence
> of instructions, which is determined by the prime numbers, causes a robot to
> look drunk, if and only if the Riemann Hypothesis is true!

~~~
j9461701
>monomer-dimer problem

Oh hey, I did my undergrad thesis on that! It generates neat looking graphics:

[https://imgur.com/a/Z6hySAw](https://imgur.com/a/Z6hySAw)

~~~
teekert
I did a Masters in Biophysics, the topic was diffusion of proteins in cell
membranes, they usually show random walks (albeit restricted to compartments
or showing distinct speeds (bound/unbound?)). That graphs does not look like a
random walk, so the Riemann Hypothesis is false?

~~~
j9461701
The graphic comes from certain assumptions that do not hold in the general
case. Specifically, that of non-isotropic fragmentation and arbitrary
fragmentation depth.

------
liquidise
Mathematics is a uniquely beautiful field to me. The commutative property has
always struck me as special in its own way. 2 x 3 = 3 x 2 feels so obvious,
but multiplication is really just addition, and 2 + 2 + 2 = 3 + 3 is far less
intuitive, yet states the very same claim.

Most fascinating to me is that many theories are effectively 1-way functions.
Entire branches of mathematics have been developed to prove otherwise
trivially stated claims. It is something to marvel at, if from a safe
distance.

~~~
ncmncm
The notion that multiplication is ("just") repeated addition is the hardest
misconception to overcome blocking progress in mathematics.

Repeated addition is an algorithm that works when you have a non-negative
integer argument. Imagining that the algorithm defines the operation limits
conception.

~~~
apalmer
I would like to understand what multiplication is in your (and i guess
advanced mathematics)

~~~
sls
The comment to which you replied says that thinking about multiplication as
"just" repeated addition is problematic, so let's look at that.

Consider 3 x 2. If we take that approach, it seems ok - we understand it to
mean "add together 3 2's" \- 2 + 2 + 2, which gives the correct answer of 6.

What about -3 * -2? What does it mean to add a negative number of times?

What about pi * pi? What does it mean to add something pi times?

What about the matrix M * the matrix N?

etc. The parent's point is that this repeated addition thing is just an
algorithm one can use to calculate a multiplication for some operands,
specifically whole numbers, not a general definition of the operation of
multiplication.

~~~
jacobolus
If we want to define _π_ × _π_ , the best we can do is some kind of algorithm
for generating an approximation to _π_ (e.g. as a continued fraction or as a
positional decimal fraction) to any desired degree, and then an algorithm for
multiplying such approximations. We can prove some bounds on the error
introduced by our multiplication algorithm, and that gives us a way of
approximating the product to any desired precision.

For example if we want to know _π_ ^2 to about 3 decimal digits (~9.87), we
can start by using an approximation of _π_ to about 4 decimal places (~3.142)
and then multiplying the two decimals.

For rational numbers, multiplication algorithms are usually built on breaking
a number down into constituent pieces, multiplying every pair of pieces from
the two multiplicands, and then adding up all of the partial products.

Matrix multiplication has the additional complication that the elementary
terms involved (entries in different places in the matrix) cannot be added to
each-other. But the basic procedure is still the same: break the two
multiplicands down into basic units which we already know a multiplication
table for, compute all of the partial products, then sum them up.

~~~
sls
I don't agree that a definition of multiplication that includes
transcendentals such as pi must be numerical. In fact, to the extent that
numerical approximations are approximations _of_ something, the thing they
would be approximating is the actual value of the operation of multiplication
being applied to pi and pi. The only reason we're talking about algorithms at
all was to distinguish between them and definitions.

~~~
jacobolus
Well, that’s a philosophical rather than mathematical question. I don’t really
believe in a concept of “actual value” outside of the context of computations
(though I don’t mind conceding it as a matter of convenience and social
convention, since the distinction almost never matters for practical
purposes). I am not an expert, but mathematicians have investigated this,
[https://en.wikipedia.org/wiki/Computable_analysis](https://en.wikipedia.org/wiki/Computable_analysis)

We can treat _π_ purely symbolically if we like, but as soon as we want to _do
anything useful_ with it we need some kind of approximation or algorithm.

~~~
sls
I am familiar with the constructivist family of ideas. Obviously I agree this
a question of philosophy, specifically the philosophy of mathematics. (So not
"rather than", since foundations of mathematics is a branch of mathematics.)
And because adopting a constructivist approach to mathematics means adopting
different ideas about what a mathematical definition _is_ or _can be_, I have
to say that it's a bit disingenuous to introduce this context only after
engaging in the above discussion.

For example, if someone asks you to explain Euclid's proof of the infinitude
of the primes, and you say that Euclid did not provide any such proof and
nothing more, I think it's quite disingenuous. It would be more proper to say,
from a constructivist view, the argument Euclid made isn't a valid proof, and
then either explain the proof in the logical context in which it was made or
decline to.

In this case, the point of discussion was separating the definition of
multiplication from an algorithm implementing it. It's quite unfair to
silently take a position that a mathematical definition without an algorithm
isn't valid or meaningful and then on that basis argue that only numerical
approximations to transcendentals have meaning.

So many common mathematical concepts such as "the integers" have no meaning in
a constructivist approach that it's not sensible to engage in mathematical
discussion without establishing that one's fundamental basis of approach
varies so widely from the common one.

------
rusanu
3Blue1Brow has a nice video on it
[https://www.youtube.com/watch?v=sD0NjbwqlYw](https://www.youtube.com/watch?v=sD0NjbwqlYw)

------
dvt
I really wish the Riemann-Zeta Function were more often explained in terms of
a prime number sieve. It's actually not particularly difficult to follow and
the connection between the function and the distribution of primes would be
completely obvious.

~~~
ncmncm
This.

In his book on the topic, William Stein did not get around to the connection
to the product of primes until page 121.

------
EpiMath
If you've got enough math background to follow it, Harold Edwards' _Riemann 's
Zeta Function_ is a gem of a book and available inexpensively from Dover.
There is an English translation of Riemann's paper at the end of the book. I
spent a worthwhile few weeks of spare time working my way through the paper (
and a lot of pencil & paper to work "between" the steps in the paper -- math
is not a spectator sport ) with a longish diversion diving into the gamma
function along the way.

~~~
sroussey
Math should be a spectator sport! ;)

------
mikorym
I think "don't try to prove the Riemann hypothesis" is only part of the
iceberg that includes "you may want to prefer theory building over problem
solving" and "we're not getting any medals here". It's interesting that this
was written by one of the category theoretic schools; the category theorists
that I studied under are quite wary of things like the RH. After all, Saunders
Mac Lane never won a Field's medal. I am not throwing shade, but it's
exceedingly difficult to try to judge (any) mathematician's "worth" in the way
a prize or medal does in popular media.

------
teh_infallible
Every now and then I try to delve into the frightening world of math. Then I
see something like this, and start to feel very tired.

Then I think, “My hair is already falling out. Do I need something like this
to accelerate the process?”

~~~
arkano
As a mathematician, I have noticed that people that like to build something
that, for example, can fly, start from a paper plane; they don't get
discouraged because they can't yet build a 737 aircraft. However, in math, you
need a lot of experience before you can even judge whether a problem is in
fact a 737 and not a paper plane (and even then, you can be mistaken). I often
see students discouraged because of this and that's why I suggest taking it
slow from the beginning.

> Every now and then I try to delve into the frightening world of math.

To want to understand is to be human. :)

~~~
khawkins
Honestly, I think the near constant exaltation of problems like the Riemann
Hypothesis, P=NP, Fermat's last theorem, etc. is more damaging to the field
than good. Many of these theorems have dubious application to anything
practical were the theorems unquestionably proven. Subsequently, it frequently
gives the impression to a lay observer that mathematics is all about number
theory and tackling pointless puzzles.

Going into undergrad I was briefly discouraged from going into mathematics
because this was the impression I got. They're interesting to think about, but
I didn't want my future to be firmly situated in inapplicable theory.

I say this knowing there is plenty of work to be done in the applied
mathematics, especially in trying to simplifying the understanding of complex
problems. I'd like to see more of the glorification of moderately hard
problems which take more time to explain but are well within the grasp of
people who start working on it, than easy to explain problems which will
likely never be in the grasp of anyone.

------
chx
If this interests you, I would strongly recommend reading
[http://www.riemannhypothesis.info/2014/10/tossing-the-
prime-...](http://www.riemannhypothesis.info/2014/10/tossing-the-prime-coin/)
this explains the relation between random walks and the RH in a surprisingly
easy to understand fashion.

------
elamje
My absolute favorite numberphile video is how pi occurs in a peculiar way with
Riemann Zeta/Mandelbrot. Maybe it only amazes me because I don’t have a PhD in
math, but it just seemed so strange how pi shows up in this video.

[https://youtu.be/d0vY0CKYhPY](https://youtu.be/d0vY0CKYhPY)

------
wmp56
Why does everybody think that by virtue of math ought to be nice, such a nice
hypothesis ought to be true? Isn't it just a form of the survivorship bias
that we observe only nice side of math? What if this hypothesis stands true
for all N < 10^10^10^467+17, and then suddenly it doesn't? Perhaps to make a
breakthru in math (and physics) we need to consider the possibility that the
reality can be ugly and counterintuitive and beyond a certain complexity
level, math and physics cannot be described by nice formulas.

~~~
BlackFingolfin
I think that's a misconception. Mathematicians are keenly aware that there are
plenty examples in history were many people "believed" (hoped? expected?) that
some result might be true because it would be "beautiful", but it turned out
to be false. Or where numerical evidence suggested something only to turn out
to be wrong in the end. And where the first counterexample only exists at
huge, numerically infeasible bound (see e.g.
[https://en.wikipedia.org/wiki/Skewes%27s_number](https://en.wikipedia.org/wiki/Skewes%27s_number))

And hence a well-known suggestion is that when tackling a hard problem, is to
try finding a proof on even days, and a counterexample on odd days...

And indeed, lots of people tried (and still try) to find counterexamples to,
or otherwise disprove, the Riemann Hypothesis. However, there are indeed many,
many results and heuristics that give a strong suggestion that the RH might be
true -- far more than mere numerical results computing zeroes of the Zeta
function. Of course none of them constitute a proof; but this really goes far
beyond a simple hope for "beauty" in the theory.

------
Yajirobe
So if RH is proven, what actually changes? As far as I know, there are tons of
theorems that already presuppose RH to be true There wouldn't suddenly be an
insight into how to find larger primes, for example.

~~~
tgb
The method used to solve RH would almost certainly contribute substantially to
our understanding of mathematics and give new directions to tackle other
questions in mathematics. For example, the resolution of the Poincare
Conjecture by Perelman involved making progress with the technique of "Ricci
flow" which is still an active area of research both in itself and for
tackling other problems. The Poincare Conjecture per se is often not involved
in these Ricci flow questions even though the proof of the conjecture is used.

------
ddxxdd
1\. I'm a big fan of John Baez.

2\. I'm getting the impression from this article that solving the Riemann
Hypothesis is similar to solving P=NP in that a solution can be used to attack
RSA encryption.

~~~
schwurb
> P=NP in that a solution can be used to attack RSA encryption.

Note that 1) P=NP does not necessarily give raise to any polynomial algorithm
that solves a NP problem. The proof would prove the existence of one such
algorithm, but it might well never be found (which is the current status quo)
2) even if it would be polynomial, it could still run longer than the heat of
the universe. O(n) = n^10000000 would still be a polynomial runtime for
example. The second reason is why Donald Knuth does think that P=NP might be
possible.

~~~
drdeca
Isn’t there some (highly impractical) algorithm which dovetails through
different Turing machines, in a way that has an asymptotically optimal runtime
for a given problem, just with really terrible constants?

I thought we knew an algorithm that, if P=NP, would solve NP problems in P
time, (but with absurd constants), and otherwise solves the problems is worse
than polytime.

But I could be remembering this totally wrong.

~~~
schwurb
> (highly impractical)

Forget everything practical - we are in deep theoretic waters here! There are
thousands of algorithm with even a polynomial solution where you still go for
the heuristic because the polynomial version is way to slow.

> Isn’t there some (highly impractical) algorithm which dovetails through
> different Turing machines, in a way that has an asymptotically optimal
> runtime for a given problem, just with really terrible constants?

Optimal might be, as long as optimal does not mean polynomial. Otherwise you
would read about it in the newspapers ;) I don't know of such algorithm, but
"trying out different turing machines" gives me a strong gut feeling of "not
polynomial".

> I thought we knew an algorithm that, if P=NP, would solve NP problems in P
> time, (but with absurd constants), and otherwise solves the problems is
> worse than polytime.

Is that the algorithm you are refering to? Sounds like what Turing proposed
once. The interesting branch is the P=NP, since then you could answer really
really interesting things in P. Theoretically - and if indeed P=NP ;)

~~~
drdeca
this doesn't say the exact algorithm (I haven't found it), but this stack
overflow answer talks about it :
[https://stackoverflow.com/questions/5107140/what-is-meant-
by...](https://stackoverflow.com/questions/5107140/what-is-meant-by-
dovetailing)

------
oneepic
I've been making a serious attempt at solving it but I'm not a mathematician.
Even still I have a few good leads yet to pursue, and I learned a ton about
the practice of mathematics that I never would've learned otherwise. (Wish I
could share my leads, but I kinda want the money and glory... :) )

The article is spot on. I've had so many moments where the math looks so fishy
that it seems like R _has_ to equal 1/2 (ie hypothesis is true), but I just
don't have the facts to prove it. In particular, it's really hard to evaluate
the infinite sums you find working thru the problem. I actually believe that
there's a good chance the hypothesis is false but we'll see someday.

~~~
monktastic1
> Wish I could share my leads, but I kinda want the money and glory... :)

Well, like Prof Baez says, unless you've solved other major open problems
before, it's probably unwise to believe you'll be the one to crack it. You are
likely to gain more by sharing your leads and seeing what mathematicians have
to say.

