The Riemann Hypothesis • Hacker News Board

Re: The Riemann Hypothesis

#11

Post by arkano » Sun, Sep 08, 2019, 8:56 PM UTC

teh_infallible wrote: post #2

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?”

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. :)

Re: The Riemann Hypothesis

#12

Post by rat9988 » Sun, Sep 08, 2019, 9:02 PM UTC

liquidise wrote: post #10

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 otherwi…

>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.

Re: The Riemann Hypothesis

#13

Post by Cybiote » Sun, Sep 08, 2019, 9:04 PM UTC

Here's a quite friendly elucidation by the inimitable Avi Wigderson, from: 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!

Re: The Riemann Hypothesis

#14

Post by ncmncm » Sun, Sep 08, 2019, 9:09 PM UTC

liquidise wrote: post #10

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 otherwi…

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.

Re: The Riemann Hypothesis

#15

Post by ncmncm » Sun, Sep 08, 2019, 9:17 PM UTC

dvt wrote: post #8

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.

This.

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

Re: The Riemann Hypothesis

#16

Post by jacobolus » Sun, Sep 08, 2019, 9:28 PM UTC

liquidise wrote: post #10

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 otherwi…

More interesting, in more general contexts multiplication is not commutative.

For example, if we apply the same pair of 3-dimensional rotations in opposite orders, we generally get different results.

Scaling and planar rotation (in combination, a.k.a. “complex numbers”) are conveniently among the types of commutative multiplication.

> multiplication is really just addition

This is a misleading summary. That multiplication of integers per se can be re-expressed as addition (or if you like, as counting) depends on the basic parts involved being very simple and uniform. The basic multiplication table for integers is just

   × | -1  0  1
  –––––––––––––
  -1 |  1  0 -1
   0 |  0  0  0
   1 | -1  0  1

Since every other integer is just some sum of these basic parts, and multiplication distributes over addition, that covers it. To multiply two integers, first break each one down into some sum of a collection of –1, 0, and 1, then look up each partial product in the basic multiplication table above, and finally sum up all of the results. This process boils down to counting. Since the basic integer multiplication table is commutative, so is integer multiplication in general.

But other kinds of numbers have richer structure based on a richer multiplication table of basic elements.

Re: The Riemann Hypothesis

#17

Post by apalmer » Sun, Sep 08, 2019, 9:30 PM UTC

ncmncm wrote: post #14

liquidise wrote: post #10

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 otherwi…

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.

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

Re: The Riemann Hypothesis

#18

Post by red75prime » Sun, Sep 08, 2019, 9:34 PM UTC

rat9988 wrote: post #12

liquidise wrote: post #10

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 otherwi…

>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.

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.

Re: The Riemann Hypothesis

#19

Post by Yajirobe » Sun, Sep 08, 2019, 9:34 PM UTC

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.

Re: The Riemann Hypothesis

#20

Post by ddxxdd » Sun, Sep 08, 2019, 9:41 PM UTC

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.