The Riemann zeta function is the function zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + .... For example, zeta(2) = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = pi^2/6. As a partially tongue-in-cheek example, zeta(-1) = 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12. Obviously it doesn't make sense to add all the positive integers (the series doesn't converge ), but if you squint and ignore this, and just do the arithmeti…
> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…
The thing impendia is referring to there is analytic continuation, which (as mentioned) is a way to extend the domain of a certain set of functions, like the zeta function. It is perfectly rigorous. His/her language was just a short-hand for "don't worry about the details here, but it does work".
Mathematicians aren't stupid, and more than any other profession, they value rigor. They know what they're doing.
The Riemann zeta function is the function zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + .... For example, zeta(2) = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = pi^2/6. As a partially tongue-in-cheek example, zeta(-1) = 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12. Obviously it doesn't make sense to add all the positive integers (the series doesn't converge ), but if you squint and ignore this, and just do the arithmeti…
> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…
But this isn't just summing up a few positive integers, in which case there's no ambiguity in what the answer is. Once you start summing up infinitely many things you have to bring in some theory and some techniques to justify what the answer is. These techniques generally have a limited scope, but there's a big theory of "divergent series" that shows that if you extend these techniques to more contexts you still get a definition that is compatible with most of what one would want from a limit. For example, taking running averages, or taking it as the coefficients of a series and taking a limit. So classically 1-1+1-1+1... doesn't converge, but if you "squint" and take the running averages of the partial sums (1,0,1,0,...) you would get 1/2. Or if you use the fact that 1+x+x^2+x^3+... = 1/(1-x) and take x=-1 you would get 1/2. Or a myriad other approaches that are perfectly valid with standard convergence, and just all happen to get you that 1-1+1-1+... is 1/2. And yes if you squint very hard you would get that 1+2+4+8+16+... = 1/(1-2)=-1
Hypothesis: As the complexity of proofs approaches the limits of human ability to understand, saying it is a proof becomes more important than proving it is a proof.
Evidence: The Wikipedia page for the Lindelöf hypothesis already unambiguously states that it has been formally proved.
The Riemann zeta function is the function zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + .... For example, zeta(2) = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = pi^2/6. As a partially tongue-in-cheek example, zeta(-1) = 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12. Obviously it doesn't make sense to add all the positive integers (the series doesn't converge ), but if you squint and ignore this, and just do the arithmeti…
> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…
In mathematics, the way to the answer is essential. If you are doing /anything/ wrong on your way, your result is void.
> Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12. I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring…
But this isn't just summing up a few positive integers, in which case there's no ambiguity in what the answer is. Once you start summing up infinitely many things you have to bring in some theory and some techniques to justify what the answer is. These techniques generally have a limited scope, but there's a big theory of "divergent series" that shows that if you extend these techniques to more contexts you still get…
The problem with this first approach is that you can get arbitrary results just be reordering the series. Also, the formula you give for the geometric series only works for |x|<1.
Hypothesis: As the complexity of proofs approaches the limits of human ability to understand, saying it is a proof becomes more important than proving it is a proof. Evidence: The Wikipedia page for the Lindelöf hypothesis already unambiguously states that it has been formally proved.
If you check the history and talk of that page you will see that there is one very persistent user who has repeatedly re-added this section while at least to others tried to remove it.
Hypothesis: As the complexity of proofs approaches the limits of human ability to understand, saying it is a proof becomes more important than proving it is a proof. Evidence: The Wikipedia page for the Lindelöf hypothesis already unambiguously states that it has been formally proved.
But this isn't just summing up a few positive integers, in which case there's no ambiguity in what the answer is. Once you start summing up infinitely many things you have to bring in some theory and some techniques to justify what the answer is. These techniques generally have a limited scope, but there's a big theory of "divergent series" that shows that if you extend these techniques to more contexts you still get…
The problem with this first approach is that you can get arbitrary results just be reordering the series. Also, the formula you give for the geometric series only works for |x|<1.
Reordering a divergent series can indeed give you any result you want. This is not reordering though, it’s called Cesàro summation. It won’t always get you an answer, but if it does then the answer is unique and reacts nicely to sums and products.
Talking about it as summation might be misleading since that’s one of those concrete terms that mathematicians like to redefine without anyone’s approval. Picture we have a library that includes many tricks and approaches for taking an infinite series as input and outputs a number. We know it works as expected on every convergent series. But we forgot to put in any preconditions and we’ve let people input things that are not convergent series. But whoa in many cases we are still getting a number out of it, and it’s always the same answer no matter what we do. Maybe that’s something worth studying?
I think it's because any information we gain about the Riemann Hypothesis (Lindelöf hypothesis is implied by RH) gives us information about the distribution of prime numbers. Any time you gain information about the distribution of prime numbers you immediately gain information that can be applied to any form of cryptography that makes use of prime numbers. You could use this information either to break existing forms…
So, people hire you to break into their places... to make sure no one can break into their places?
A Sneakers quote. Worth watching for Hollywood's take on the early 90s version of our current social media/social engineering quagmire.