I'm reading this book called the Music of Primes and the question has to do with the Riemann Zeta function which is related to prime numbers. By looking at wikipedia I am in way over my head in terms of the mathematics, but I am trying to figure it out. Why does the Zeta function output -1/2 at 0 and -1/12 at -1? In other words why does 1+1+1+1+...= -1/2? And why does 1+2+3+4+...= -1/12?
off the top of my head, I think it has something to do with series properties. Geometric series, and infinite series do converge to fractional answers.
For why 1+1+1+1+1...= -1/2: http://planetmath.org/encyclopedia/ValueOfTheRiemannZetaFunctionAtS0.html I think the point is that if you do it by the sum, you are doing it in a discrete fashion, but s is continuous at s=0 so you need to integrate it instead. My best guess.
the key is understanding that it is a meromorphic function on the whole complex s-plane. once you get that, the rest is cake.
I've got a lot to learn! So much terminology that I have no clue about. It's amazing I even managed get to differential equations (M427K) without even really understanding calculus, logs, pi, e, trig functions, or almost anything in math. I really wish I would have learned the conceptual side of mathematics rather than just trying to memorize patterns and equations. That way of going thru school ultimately resulted in a colossal collapse at UT. It's pretty daunting looking at all this stuff and recognizing that I actually studied the foundations of what these things are based off of for years, but never really learned anything. As a result I have almost no intuitive feel for concepts that I should since I never really understood them in the first place. It's hard to even figure out where to begin after seeing how many holes I have in my math foundation. ...logs into KhanAcademy.org
well once you learn what a meromorphic function is, let me know because i have no idea. i just wish the riemann zeta function wasn't seemingly in every wikipedia article on obscure math/number theory concepts i get to through hyperlinks.
I have a math degree and got one more semester till getting a masters, and this stuff is over my head. Too many functions. I can't seem to define it. I know this: 1+1+1+1... does not equal -1/2. It equals infinity. Some weird property of the function makes it equal -1/2. Same with the 1+2+3+4... thing.
All I can gather is that there is some sort of regularization technique going on here...and I don't even really know what that is http://en.wikipedia.org/wiki/Zeta_function_regularization#Definition
