|Euler's Zeta Function|
Leonhard Euler (1707-1783) was a Swiss mathematician who played an important role in the development of mathematics. One of his remarkable results was the following expression for the so-called zeta function.
Where s > 1, n are the positive integer numbers (1,2,3,..., etc), and p are the prime numbers (2,3,5,7,11,13,...,etc). Told you only geeks would appreciate this. Anyway, as an example, if we choose s = 2, and write out a few terms we find that:
1 + 1/4 + 1/9 + 1/16 + 1/25 +... = (1/(1-(1/4)) * (1/(1-(1/9)) * (1/(1-1/25))*...
Doesn't really matter. It's just a way to express an infinite sum of expressions of whole numbers as an infinite product of expressions of prime numbers.
What??? What do whole numbers have to do with prime numbers? How do they know about one another? Why should there be any relationship whatsoever between them?
Euler's result totally blew me away when I first read about it. To me, this raises philosophical questions that resemble: "If a tree falls down in a forest, but there is nobody around to hear it, did the tree really fall down?". Or: "If man did not exist, would prime numbers exist?". I guess the answer to the latter question must be: "Yes, if whole numbers exist, then prime numbers must exist".
Call me crazy, but I find the whole thing amazing.
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