20 ms·

The Riemann Hypothesis

294 points by gballan 7y ago

teh_infallible 7y ago
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?”
- HankB99 7y ago
  I'm satisfied to know that there are people who need something like this.
- v64 7y ago
  For what it's worth, most mathematicians find the Riemann hypothesis frightening as well. There are much lower level aspects of math to be explored without climbing the proverbial Mt. Everest of math problems.
  - mlthoughts2018 7y ago
    I disagree with this. I don’t think the Riemann Hypothesis is “harder” than various other unsolved problems like Yang-Mills or Navier-Stokes. Most professional mathematicians would succeed just as much or as little as the particular subset who actively work on the Riemann Hypothesis, if they chose to study it. I think it’s more that in terms of instrumental rationality / goal achievement, working on the Riemann Hypothesis doesn’t suit very many people. There are other trade-offs with funding, applications, lifestyle goals, etc., so it’s just not for everyone. But there’s no way at all that “most mathematicians find the Riemann Hypothesis frightening” as you suggest.
    - lonelappde 7y ago
      The other unsolved problems at frightening as well, insofar as such an informal emotional word has any consensus meaning, perhaps "if I had funding to spend my whole life on this problem, there's a god chance I'd make very little progress."
      - v64 7y ago
        Agreed, I think frightening carried a connotation that I didn't intend. My intention was to state that problems like the Riemann hypothesis are of a level of difficulty such that mathematicians believe these problems cannot be solved without a major fundamental leap forward in our understanding. Erdos expressed a similar frustration toward the Collatz conjecture when he stated "Mathematics may not be ready for such problems". So with respect to the original comment I was replying to, there are other areas of mathematics that one can study that are well understood and not so opaque as the Riemann hypothesis.
- atemerev 7y ago
  Mathematicians usually have more or less okay hair. Something to do with the blood flow from all that thinking.
- arkano 7y ago
  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 7y ago
    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.
  - teh_infallible 7y ago
    Thank you for your elegant reply. I really appreciated it because I am a non-engineer with a 3d printer, so your paper plane example was quite relevant. “Tinkering” with mathematical concepts is fundamentally different from the kind of tinkering I can do, but looking from the outside in.. I’ll probably pick up a calculus textbook one day :)
rusanu 7y ago
3Blue1Brow has a nice video on it https://www.youtube.com/watch?v=sD0NjbwqlYw https://www.youtube.com/watch?v=sD0NjbwqlYw
EpiMath 7y ago
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 7y ago
  Math should be a spectator sport! ;)
dvt 7y ago
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 7y ago
  This. In his book on the topic, William Stein did not get around to the connection to the product of primes until page 121.
liquidise 7y ago
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.
- rat9988 7y ago
  >2 x 3 = 3 x 2 feels so obvious You just got used to it because you learned it very young. I'm pretty sure it wasn't that obvious when you learned it.
  - red75prime 7y ago
    If I remember correctly, in my school multiplication was explained in terms of rectangles made of unit squares. Commutativity was literally visible to the naked eye then.
    - quickthrower2 7y ago
      I also think you define integers as successors. Z, S(Z), S(S(Z)) etc. you can prove that a * b = b * a, given that you have defined the + function and * functions.
    - lonelappde 7y ago
      It's both intuitively obvious geometrically, and algebraically is conceptually tricky to grasp. Or, more precisely it's trivial to decide that arithmetic operations are commutative or not, but tricky to decide correctly for all operations a child knows.
  - r_c_a_d 7y ago
    When my kids started school I was surprised to learn that at age 4 it is not obvious that the number of items on a whiteboard remains constant if all the items are moved to a different part of the board. So what seems obvious, probably is something we just don't remember learning.
Cybiote 7y ago
Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: https://www.ias.edu/ideas/2009/wigderson-randomness-pseudorandomness https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora... > 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 7y ago
  >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
  - jamiek88 7y ago
    Looks like an X-ray pic of a system on a chip.
  - unixhero 7y ago
    Something's going in there.
  - teekert 7y ago
Yajirobe 7y ago
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.
- schwurb 7y ago
  It's the same story as with many other deep and important math problems: Practically, not much would change. Many papers already assume RH - we would just have confirmation that they are reality then. What would change is that we would likely have novel and powerful methods that were used in the proof. This is what most of the buzz is about. For example, the proof of Fermats Last Theorem introduced novel connections between sub fields of mathematics.
- tgb 7y ago
  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.
- m11a 7y ago
  Back in the 20th century there was a goal to ensure maths was consistent, notably by David Hilbert, and that everything could be proven from a finite set of axioms, and he wanted to ensure everything was built on a logical foundation and was rigorous. In practice it may not change much, but more philosophically speaking, it's dangerous to take something as granted without a proof. If we then have many ideas building off this unproven idea, and it's later shown to be false, a lot of those theorems would be based off a false premise. This doesn't necessarily mean they the proposition is incorrect, but it would mean that they are unproven. It's more to satisfy the rigour in maths, and also pose as a challenge to mathematicians. In the journey to proving RH a lot of new ideas could be found that could prove useful, too.
- 7y ago
ddxxdd 7y ago
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 7y ago
  > 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 7y ago
    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 7y ago
      > (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 ;)
oneepic 7y ago
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 7y ago
  > 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.
- lonelappde 7y ago
  Since you have no money or glory in math yet, your better bet is to cash in your leads so you end your life with at least one bit. Impressively, almost every sentence in your comment was already addressed by the OP linked Twitter thread of warnings.
sarajames657 7y ago
solve this https://twitter.com/LADYGAGAALBUM/status/1170822977228857344 https://twitter.com/LADYGAGAALBUM/status/1170822977228857344
chx 7y ago
If this interests you, I would strongly recommend reading http://www.riemannhypothesis.info/2014/10/tossing-the-prime-coin/ http://www.riemannhypothesis.info/2014/10/tossing-the-prime-... this explains the relation between random walks and the RH in a surprisingly easy to understand fashion.
mikorym 7y ago
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.
elamje 7y ago
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
deleted 7y ago
[deleted]
wmp56 7y ago
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.
- ncmncm 7y ago
  If it's ugly, mathematicians lose interest. So, only the beautiful stuff remains in the set of things taught by mathematicians. There could be beautiful stuff concealed by a layer of ugly than mathematics never breaches.
- BlackFingolfin 7y ago
  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.

294 points by gballan 7y ago

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?”

HankB99 7y ago
I'm satisfied to know that there are people who need something like this.
v64 7y ago
For what it's worth, most mathematicians find the Riemann hypothesis frightening as well. There are much lower level aspects of math to be explored without climbing the proverbial Mt. Everest of math problems.
- mlthoughts2018 7y ago
  I disagree with this. I don’t think the Riemann Hypothesis is “harder” than various other unsolved problems like Yang-Mills or Navier-Stokes. Most professional mathematicians would succeed just as much or as little as the particular subset who actively work on the Riemann Hypothesis, if they chose to study it. I think it’s more that in terms of instrumental rationality / goal achievement, working on the Riemann Hypothesis doesn’t suit very many people. There are other trade-offs with funding, applications, lifestyle goals, etc., so it’s just not for everyone. But there’s no way at all that “most mathematicians find the Riemann Hypothesis frightening” as you suggest.
  - lonelappde 7y ago
    The other unsolved problems at frightening as well, insofar as such an informal emotional word has any consensus meaning, perhaps "if I had funding to spend my whole life on this problem, there's a god chance I'd make very little progress."
    - v64 7y ago
      Agreed, I think frightening carried a connotation that I didn't intend. My intention was to state that problems like the Riemann hypothesis are of a level of difficulty such that mathematicians believe these problems cannot be solved without a major fundamental leap forward in our understanding. Erdos expressed a similar frustration toward the Collatz conjecture when he stated "Mathematics may not be ready for such problems". So with respect to the original comment I was replying to, there are other areas of mathematics that one can study that are well understood and not so opaque as the Riemann hypothesis.
atemerev 7y ago
Mathematicians usually have more or less okay hair. Something to do with the blood flow from all that thinking.
arkano 7y ago
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 7y ago
  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.
- teh_infallible 7y ago
  Thank you for your elegant reply. I really appreciated it because I am a non-engineer with a 3d printer, so your paper plane example was quite relevant. “Tinkering” with mathematical concepts is fundamentally different from the kind of tinkering I can do, but looking from the outside in.. I’ll probably pick up a calculus textbook one day :)

rusanu 7y ago

3Blue1Brow has a nice video on it https://www.youtube.com/watch?v=sD0NjbwqlYw https://www.youtube.com/watch?v=sD0NjbwqlYw

EpiMath 7y ago

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 7y ago
Math should be a spectator sport! ;)

dvt 7y ago

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 7y ago
This. In his book on the topic, William Stein did not get around to the connection to the product of primes until page 121.

liquidise 7y ago

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.

rat9988 7y ago
>2 x 3 = 3 x 2 feels so obvious You just got used to it because you learned it very young. I'm pretty sure it wasn't that obvious when you learned it.
- red75prime 7y ago
  If I remember correctly, in my school multiplication was explained in terms of rectangles made of unit squares. Commutativity was literally visible to the naked eye then.
  - quickthrower2 7y ago
    I also think you define integers as successors. Z, S(Z), S(S(Z)) etc. you can prove that a * b = b * a, given that you have defined the + function and * functions.
  - lonelappde 7y ago
    It's both intuitively obvious geometrically, and algebraically is conceptually tricky to grasp. Or, more precisely it's trivial to decide that arithmetic operations are commutative or not, but tricky to decide correctly for all operations a child knows.
- r_c_a_d 7y ago
  When my kids started school I was surprised to learn that at age 4 it is not obvious that the number of items on a whiteboard remains constant if all the items are moved to a different part of the board. So what seems obvious, probably is something we just don't remember learning.

Cybiote 7y ago

Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: https://www.ias.edu/ideas/2009/wigderson-randomness-pseudorandomness https://www.ias.edu/ideas/2009/wigderson-randomness-pseudora... > 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 7y ago
>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
- jamiek88 7y ago
  Looks like an X-ray pic of a system on a chip.
- unixhero 7y ago
  Something's going in there.
- teekert 7y ago

Yajirobe 7y ago

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.

schwurb 7y ago
It's the same story as with many other deep and important math problems: Practically, not much would change. Many papers already assume RH - we would just have confirmation that they are reality then. What would change is that we would likely have novel and powerful methods that were used in the proof. This is what most of the buzz is about. For example, the proof of Fermats Last Theorem introduced novel connections between sub fields of mathematics.
tgb 7y ago
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.
m11a 7y ago
Back in the 20th century there was a goal to ensure maths was consistent, notably by David Hilbert, and that everything could be proven from a finite set of axioms, and he wanted to ensure everything was built on a logical foundation and was rigorous. In practice it may not change much, but more philosophically speaking, it's dangerous to take something as granted without a proof. If we then have many ideas building off this unproven idea, and it's later shown to be false, a lot of those theorems would be based off a false premise. This doesn't necessarily mean they the proposition is incorrect, but it would mean that they are unproven. It's more to satisfy the rigour in maths, and also pose as a challenge to mathematicians. In the journey to proving RH a lot of new ideas could be found that could prove useful, too.
7y ago

ddxxdd 7y ago

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 7y ago
> 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 7y ago
  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 7y ago
    > (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 ;)

oneepic 7y ago

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 7y ago
> 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.
lonelappde 7y ago
Since you have no money or glory in math yet, your better bet is to cash in your leads so you end your life with at least one bit. Impressively, almost every sentence in your comment was already addressed by the OP linked Twitter thread of warnings.

sarajames657 7y ago

solve this https://twitter.com/LADYGAGAALBUM/status/1170822977228857344 https://twitter.com/LADYGAGAALBUM/status/1170822977228857344

chx 7y ago

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

mikorym 7y ago

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.

elamje 7y ago

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

deleted 7y ago

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wmp56 7y ago

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.

ncmncm 7y ago
If it's ugly, mathematicians lose interest. So, only the beautiful stuff remains in the set of things taught by mathematicians. There could be beautiful stuff concealed by a layer of ugly than mathematics never breaches.
BlackFingolfin 7y ago
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.