– Can anyone describe Riemann hypothesis? You can ELI15 also. I am okay with it. I know it’s kind of impossible but still asking the question. So any mathematician here.
In: Mathematics
I would strongly recommend a video like this:
As it’s very hard to understand complex numbers without a visual of the complex plane. Maybe ask again if you have questions after seeing the video?
Ok, so I’ll try my best, Riemann hypothesis is a conjecture, this means, a mathematician thought something was true but really has no idea how to prove that it is actually true, it sounds like a simply I don’t know kind of thing but conjectures are important not only cause they provide a goal for which mathematics can grow but also this are very well thought questions with huge implications in a variety of branches.
The conjecture has to do with a particular function that is complex valued this means instead of being a function that turns a number into another number it turns a complex number into another complex number, I’m assuming you understand what a functions is, in case you don’t, simply put a functions is a kind of machine that turns some thing into other thing and it associates the first thing (it’s input) with the second (it’s value) for more information you can search online is not a hard thing to grasp and you only need the basics of it.
Then you need to understand (on a basic level) what a complex number is, you can picture a number in the number line as a single dot in that number line, now imagine that instead of a line we have a plane so, every single dot on the plane is a number, this numbers are the complex numbers, this is a oversimplified description, I’ll advice to seek for more information online on the topic as it is very deep and interesting.
Back to the track, this function is called a Rieman zeta function (zeta is a greek letter), this function is written on the terms of a sum, is basically an infinite sum that (sometimes) converges to a number, this number is the output of the function.
Now the problem, this function is incomplete, well you see if you have a very smart calculator or some math program you can ask for the value of the riemman zeta function when you input lets say “2” and it will probably give you a good aproximation, you can even input complex numbers that look like “3+5i” and it will give you and output too.
But if you try to input “5/7 + 3i” the computer will just scream at you, if the first number (the real part) of the complex number is greater than one then the function is defined (we figured it out) also some other numbers like negative integers or 1/2 and others are also defined but everything else is just a big I dunno.
And to make matters more interesting, this functions pops up in different places of math, probably you have heard about the relation with prime numbers, the “Euler product formula” is an equality that very strongly relates prime numbers with values of this function, so understanding how this funtion behave can lead us to understand more about prime numbers, this kind of numbers and their behavior are extremly important not only for the sake of math but also for the sake of our current technology (the entire internet security protocol is made on top of them).
So we want to complete this this function ASAP the problem is we have no idea how to do it, and it appears to be really hard (maybe impossible?(like mathematically not just humanly)) so a great leverage point will be to know all the places where the function outputs a cero.
The hypothesis conjectures that ALL the non-trivial (This means not obvious and actually usefull) zeroes have a real part (the first number of a complex number) equal to 1/2.
If this is true it will lead to a lot of things, actually a lot of people have actually take the bet, assume it’s true and probe a lot of things with it, on the converse some people works as if it was false, we are not really fighting about it, we are just pursuing the truth.
Acutally what interest us is not really if the conjucture is true or not, but the actual proof of it, cause it will lead to a lot of new math.
TLDR:
Really weird function that is incomplete and is very important, we want to complete it and a good start point is to know the zeroes of the functions, conjecture says all of the usefull zeroes have a real part of 1/2
PD:
Sorry for my grammar if I made some mistake, my english is a little lacking. Hope it helped!
First, let’s talk a bit about the motivation. Prime numbers are really important in math. They’re the building blocks that make up all the other whole numbers. A natural question you might ask is, “How many prime numbers are there?” A few thousand years ago, the Greeks figured out there are infinitely many. The next question you might ask is, “How many are there below 100? Or 1000? Or 1000000? Or more generally, how many primes are there below x?”
The answer, it turns out, is pretty close (in a very precise sense) to x/log(x). And we can do better than that. If you play some tricks with integrals, you can come up with the Li(x) function, which is an even better approximation. If we play with these approximations, a certain function, known as the Riemann zeta function, shows up a lot. And so to really understand the prime numbers, we want to understand the zeta function.
One of the best ways to understand a function is to know where it evaluates to 0. It’s not too hard to show zeta(x) is 0 when x is a negative even number. But if we allow for complex numbers as inputs, there are other places where zeta(x) is 0. In particular, the Riemann Hypothesis says that all of those are complex numbers with real part 1/2.
Latest Answers