The Riemann Zeta Function
Here is a view of the Riemann Zeta function graphed from x=1.2 to 10. You will notice a sharp spike as x goes toward 1, where it shoots off to infinity. The Riemann Zeta function at x=1 is the harmonic series. Since *everybody* knows the harmonic series diverges, so does the Riemann Zeta function at x=1. As x gets larger, the function approaches 1 quickly. This function directly determines the statistical properties of the distribution of prime numbers, so mathematician go wild studying everything about it.
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