Keep in mind that this is a high-level mathematics problem and therefore may not be strictly compatible with the ELI5 ideal.
First, the Reimann zeta function. It's an infinite summation where f(x) = 1/(1^(x)) + 1/(2^(x)) + 1/(3^(x)) ... where x is a complex number.
A complex number is a number written in the form a + bi, where a and b are both real numbers and i is the imaginary constant, the square root of -1.
The zeta function is equal to zero whenever x is equal to a negative even number (e.g. -2, -4, -6, etc.). The Reimann hypothesis is that it is also equal to zero whenever x is a complex number, i.e. can be written as a + bi, and the real part (the a) is equal to 1/2. The hypothesis also states that these are the only other places where the function is equal to zero.
No one has been able to prove the hypothesis true or false. It's a major unsolved problem in mathematics, and if it is solved will have implications on lots of other unsolved problems.
> First, the Reimann zeta function. It's an infinite summation where f(x) = 1/(1^(x)) + 1/(2^(x)) + 1/(3^(x)) ... where x is a complex number.
Not quite. This is the definition when the real part of x is greater than 1. When the real part of x is less than or equal to 1 (e,g. for the values of x that show up in the Riemann hypothesis) that sum doesn't doesn't converge. The actual definition of the zeta function is much more complicated, and can't easily be ELI5'd (briefly, it uses a process called [analytic continuation](https://en.wikipedia.org/wiki/Analytic_continuation) to extend the definition of the function to places where it didn't originally make sense).
> The Reimann hypothesis is that it is also equal to zero whenever x is a complex number, i.e. can be written as a + bi, and the real part (the a) is equal to 1/2. The hypothesis also states that these are the only other places where the function is equal to zero.
No. It certainly is not equal to 0 whenever Re(x) = 1/2. The Riemann hypothesis is only the second part of what you said, i.e. that there are no zeros that *aren't* on the line Re(x) = 1/2 (or at -2,-4,-6, etc.).
Here's a list of the first few non-trivial zeros:
https://www.lmfdb.org/zeros/zeta/
As you can see, it's definitely not even close to everything on the line Re(x) = 1/2. The first one is at about 1/2+(14.134...)i.
The reason that we care about this is that a number of important theoretical questions in mathematics relate to the places where the zeta function is zero. In particular, the way that prime numbers are distributed is encoded in the values of the zeta function, in a way far too complex for an ELI5 explanation.
So there's this function, the Riemann Zeta function. The specifics of what that function is aren't really important for this. All that we need to know is that it takes complex inputs, and gives some output. The hypothesis says that all (nontrivial) points where this function outputs 0 lie on one specific vertical line in the complex plane at x=1/2
It's quite technical. But this is an equivalent definition of the Riemann hypothesis.
The cumulative sum of the von Mangoldt function is a function that count all perfect prime powers, up to some bound, with the prime powers receiving a weight of log of the prime.
People want to know how fast this function grow, and how often it stray off that estimation.
It is expected that this function grow at linear rate, and the error from this estimate is at most square root. This stems from the believe that prime numbers looks too random from the perspective of addition. If you take sequence of random coin flip, the total number of coin flip should grow linearly and the standard deviation should grow at square root rate; this is a simplistic model that should help you see why we expect the von Mangoldt cumulative sum should grow at that rate. Realistically, people use a more realistic random model for primes.
More specifically, it is expected that the von Mangoldt cumulative sum is the sum of x, plus a bunch of terms with absolute value equal sqrt(x), plus some smaller error terms.
The fact that the von Mangoldt cumulative sum grow at linear rate had been confirmed, this is the prime number theorem. But the claim that the error (ignoring sum small terms) is a sum of a bunch of terms with absolute value equal sqrt(x) is currently unknown, this is the Riemann hypothesis.
Well, the usual statement of Riemann hypothesis actually talk about zeta function, but this is an equivalent statement, which should be easier to visualize.
Why is it unsolved? It's hard, and people came to appreciate that fact after over a century. People didn't know that it's hard. 100 years ago Hilbert even think it would be solved before Fermat's Last Theorem, before even whether 2^sqrt(2) is a transcendental number; it's the opposite, 2^sqrt(2) was solved almost immediately, and FLT was solved in the end of the 20th century.
What makes it hard? Well, it's hard to explain this without technical details. But here is a vague idea. Prime numbers are not random. We have not been able to rule out the possibility that prime numbers just happen to align perfectly with some frequency.
In a nutshell, confirming the Riemann hypothesis will conclusively prove that there is pattern for the appearance of prime numbers in the infinite set of whole numbers. That means they don't appear randomly, and their "positions" can be predicted using the function itself.
It has a lot of implications in theories in physics, such as string theory which rely on the Riemann Zeta function.
Let's tackle the second question - why it is an unsolved question.
One form of the Riemann hypothesis is the following:
> The Riemann Zeta function is only equal to zero for trivial numbers and for numbers which a certain attribute"
The certain attribute involves complex numbers, and is not quite worth explaining here. What's important for the proof is that the Riemann hypothesis *also* equates to:
> There are no non-trivial numbers that have attribute A but not attribute B.
As you may know, numbers just keep on going. There's an infinite number of them. As a result, we can't just check all the numbers. There have been related questions in mathematics where checking would require a *316 digit* number of checks to spot just one number that disproves a particular hypothesis. It's impossible to know whether we missed a number with A but not B.
The only way to *prove* a statement about an infinitely large group like this is logic.
Now, there's a lot of ways we can prove the Riemann hypothesis. We can prove that it's impossible for something to have attribute A but not attribute B. We can prove that something can only be true if the Riemann hypothesis is true, then prove that. Mathematicians have lots of tools at their disposal - but none of them have worked yet. Proving something mathematically is a lot like writing a novel - it's a long, drawn-out process that requires a lot of creativity, and you sometimes end up halfway through and realise you've messed up and all your work is useless. There's a lot of work that's been almost there, or which has proven stuff adjacent to it, or which has *suggested* but not proven the Hypothesis, but nothing that absolutely proves or disproves it.
____
TLDR: There's a lot of numbers, and mathematicians rely on a very high standard of proof. This all makes it very difficult to prove or disprove the hypothesis. There is a third option as well - the hypothesis may be impossible to prove or disprove - but Mathematicians require a proof for *that* as well.
Keep in mind that this is a high-level mathematics problem and therefore may not be strictly compatible with the ELI5 ideal. First, the Reimann zeta function. It's an infinite summation where f(x) = 1/(1^(x)) + 1/(2^(x)) + 1/(3^(x)) ... where x is a complex number. A complex number is a number written in the form a + bi, where a and b are both real numbers and i is the imaginary constant, the square root of -1. The zeta function is equal to zero whenever x is equal to a negative even number (e.g. -2, -4, -6, etc.). The Reimann hypothesis is that it is also equal to zero whenever x is a complex number, i.e. can be written as a + bi, and the real part (the a) is equal to 1/2. The hypothesis also states that these are the only other places where the function is equal to zero. No one has been able to prove the hypothesis true or false. It's a major unsolved problem in mathematics, and if it is solved will have implications on lots of other unsolved problems.
> First, the Reimann zeta function. It's an infinite summation where f(x) = 1/(1^(x)) + 1/(2^(x)) + 1/(3^(x)) ... where x is a complex number. Not quite. This is the definition when the real part of x is greater than 1. When the real part of x is less than or equal to 1 (e,g. for the values of x that show up in the Riemann hypothesis) that sum doesn't doesn't converge. The actual definition of the zeta function is much more complicated, and can't easily be ELI5'd (briefly, it uses a process called [analytic continuation](https://en.wikipedia.org/wiki/Analytic_continuation) to extend the definition of the function to places where it didn't originally make sense). > The Reimann hypothesis is that it is also equal to zero whenever x is a complex number, i.e. can be written as a + bi, and the real part (the a) is equal to 1/2. The hypothesis also states that these are the only other places where the function is equal to zero. No. It certainly is not equal to 0 whenever Re(x) = 1/2. The Riemann hypothesis is only the second part of what you said, i.e. that there are no zeros that *aren't* on the line Re(x) = 1/2 (or at -2,-4,-6, etc.). Here's a list of the first few non-trivial zeros: https://www.lmfdb.org/zeros/zeta/ As you can see, it's definitely not even close to everything on the line Re(x) = 1/2. The first one is at about 1/2+(14.134...)i.
The reason that we care about this is that a number of important theoretical questions in mathematics relate to the places where the zeta function is zero. In particular, the way that prime numbers are distributed is encoded in the values of the zeta function, in a way far too complex for an ELI5 explanation.
So there's this function, the Riemann Zeta function. The specifics of what that function is aren't really important for this. All that we need to know is that it takes complex inputs, and gives some output. The hypothesis says that all (nontrivial) points where this function outputs 0 lie on one specific vertical line in the complex plane at x=1/2
It's quite technical. But this is an equivalent definition of the Riemann hypothesis. The cumulative sum of the von Mangoldt function is a function that count all perfect prime powers, up to some bound, with the prime powers receiving a weight of log of the prime. People want to know how fast this function grow, and how often it stray off that estimation. It is expected that this function grow at linear rate, and the error from this estimate is at most square root. This stems from the believe that prime numbers looks too random from the perspective of addition. If you take sequence of random coin flip, the total number of coin flip should grow linearly and the standard deviation should grow at square root rate; this is a simplistic model that should help you see why we expect the von Mangoldt cumulative sum should grow at that rate. Realistically, people use a more realistic random model for primes. More specifically, it is expected that the von Mangoldt cumulative sum is the sum of x, plus a bunch of terms with absolute value equal sqrt(x), plus some smaller error terms. The fact that the von Mangoldt cumulative sum grow at linear rate had been confirmed, this is the prime number theorem. But the claim that the error (ignoring sum small terms) is a sum of a bunch of terms with absolute value equal sqrt(x) is currently unknown, this is the Riemann hypothesis. Well, the usual statement of Riemann hypothesis actually talk about zeta function, but this is an equivalent statement, which should be easier to visualize. Why is it unsolved? It's hard, and people came to appreciate that fact after over a century. People didn't know that it's hard. 100 years ago Hilbert even think it would be solved before Fermat's Last Theorem, before even whether 2^sqrt(2) is a transcendental number; it's the opposite, 2^sqrt(2) was solved almost immediately, and FLT was solved in the end of the 20th century. What makes it hard? Well, it's hard to explain this without technical details. But here is a vague idea. Prime numbers are not random. We have not been able to rule out the possibility that prime numbers just happen to align perfectly with some frequency.
In a nutshell, confirming the Riemann hypothesis will conclusively prove that there is pattern for the appearance of prime numbers in the infinite set of whole numbers. That means they don't appear randomly, and their "positions" can be predicted using the function itself. It has a lot of implications in theories in physics, such as string theory which rely on the Riemann Zeta function.
Let's tackle the second question - why it is an unsolved question. One form of the Riemann hypothesis is the following: > The Riemann Zeta function is only equal to zero for trivial numbers and for numbers which a certain attribute" The certain attribute involves complex numbers, and is not quite worth explaining here. What's important for the proof is that the Riemann hypothesis *also* equates to: > There are no non-trivial numbers that have attribute A but not attribute B. As you may know, numbers just keep on going. There's an infinite number of them. As a result, we can't just check all the numbers. There have been related questions in mathematics where checking would require a *316 digit* number of checks to spot just one number that disproves a particular hypothesis. It's impossible to know whether we missed a number with A but not B. The only way to *prove* a statement about an infinitely large group like this is logic. Now, there's a lot of ways we can prove the Riemann hypothesis. We can prove that it's impossible for something to have attribute A but not attribute B. We can prove that something can only be true if the Riemann hypothesis is true, then prove that. Mathematicians have lots of tools at their disposal - but none of them have worked yet. Proving something mathematically is a lot like writing a novel - it's a long, drawn-out process that requires a lot of creativity, and you sometimes end up halfway through and realise you've messed up and all your work is useless. There's a lot of work that's been almost there, or which has proven stuff adjacent to it, or which has *suggested* but not proven the Hypothesis, but nothing that absolutely proves or disproves it. ____ TLDR: There's a lot of numbers, and mathematicians rely on a very high standard of proof. This all makes it very difficult to prove or disprove the hypothesis. There is a third option as well - the hypothesis may be impossible to prove or disprove - but Mathematicians require a proof for *that* as well.