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The Riemann Hypothesis

golem.ph.utexas.edu

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Re: The Riemann Hypothesis

#31

Post by quickthrower2 »

red75prime wrote: post #18
rat9988 wrote: post #12

Earlier quoted context omitted.

>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.

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.

Re: The Riemann Hypothesis

#32

Post by james_s_tayler »

apalmer wrote: post #17
ncmncm wrote: post #14

Earlier quoted context omitted.

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)

For that you want to check out Group Theory.

Re: The Riemann Hypothesis

#33

Post by jacobolus »

sls wrote: post #28
apalmer wrote: post #17

Earlier quoted context omitted.

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

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…

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.

Re: The Riemann Hypothesis

#34

Post by tgb »

Yajirobe wrote: post #19

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.

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.

Re: The Riemann Hypothesis

#35

Post by lonelappde »

red75prime wrote: post #18
rat9988 wrote: post #12

Earlier quoted context omitted.

>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.

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.

Re: The Riemann Hypothesis

#36

Post by sroussey »

EpiMath wrote: post #6

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 long…

Math should be a spectator sport! ;)

Re: The Riemann Hypothesis

#37

Post by lonelappde »

ddxxdd wrote: post #20

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.

No, they don't give rise to attacks because the lower bound of NP-complete algorithms) is monstrously huge.

People tend to treat P as Omega(n^4) because all the P algorithms they know are. But that's not because P is everywhere that small, it's because larger P algorithms are just as infeasible as O(e^n) in practical computers will ever be for human timespans, as long as you choose a reasonably large N, so they aren't studied much in practice.

Re: The Riemann Hypothesis

#38

Post by lonelappde »

oneepic wrote: post #24

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 h…

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.

Re: The Riemann Hypothesis

#39

Post by scythe »

apalmer wrote: post #17
ncmncm wrote: post #14

Earlier quoted context omitted.

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)

Multiplication can have a number of meanings, but there are two important ones:

- multiplication as function composition, eg the product of two matrices is the linear transform obtained by applying the right matrix and then the left (onto a column vector)

- multiplication as “the operation that distributes over addition” — in the theory of rings and algebras, if you have one operator “+” which forms an abelian group, then any operator “•” such that (a+b)•(c+d) = ac+ad+bc+bd for all a,b,c,d then you have a ring (if • is associative) or an algebra (otherwise) and you can apply all kinds of structure theorems.

This is very oversimplified but it’s important to remember that you can multiply lots of things which don’t look like integers at all.

Re: The Riemann Hypothesis

#40

Post by lonelappde »

mlthoughts2018 wrote: post #7
v64 wrote: post #4

Earlier quoted context omitted.

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.

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 Hypot…

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."
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