Riemann Zeta Function
Imagine a sum that keeps getting bigger: 1 + 2 + 3 + 4 + ... but somehow, its "value" is said to be \[1+2+3+4+5… = -\frac{1}{12}\] This equation isn’t a joke. It isn’t a typo. And it definitely isn’t like one of those internet tricks that try to prove that 2 = 3. But how? Doesn’t calculus dictate that the sum should diverge? How can an infinite sum of positive integers result in a negative fraction? This equation is one of the paradoxical results of the famed Riemann Zeta function: \[\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\] First Encounters My first encounter of the Riemann Zeta function was in a problem introduced to me in Math circle India:\(\newline\) What is the probability that two random numbers x & y are coprime i.e. gcd(x,y) = 1 for \( x,y \in \mathbb N\) At first glance, it seems like an impossible question to compute. We’re picking 2 numbers at random — how do you even define a probability over an...
