euler's formula

For a real number s, a positive integer n, and a prime number p, we have the astonishing result that

∑_n 1/n^s = ∏_p p^s/(p^s − 1)

That is, a sum over the integers is equal to a product over the primes. This result, attributed to the prolific Swiss mathematician Leonhard Euler [1707-1783], is crucial to much of the analysis and number theory that has followed.

Note: The product is sometimes written as ∏_p 1/(1 − 1/p^s).

Proof: Observe that both the product and the sum converge for all s > 1.

Next, recall the identity q/(q − 1) = 1 + 1/q + 1/q² + ...

So we have p^s/(p^s − 1) = ∑_n 1/p^ns, where the sum is over the nonnegative integers.

Expanding the right side of the identity, we have

∑_n 1/p^ns = 1 + 1/p^s + 1/p^2s + 1/p^3s + ...

It follows that

∏_p ∑_n 1/p^ns = (1 + 1/2^s + 1/2^2s + 1/2^3s + ...)(1 + 1/3^s + 1/3^2s + 1/3^3s + ...)(1 + 1/5^s + 1/5^2s + 1/5^3s + ...) ...

Such an infinite product is unwieldy, to be sure, but we observe that a general term in its expansion is of the form 1/2^as·3^bs·5^cs·... = 1/(2^a·3^b·5^c·...)^s, where a, b, c, ... are non-negative integers.

We now use the Fundamental Theorem of Arithmetic, which says that we can uniquely express each integer as a product of primes. That is, there is a one-to-one correspondence between the positive integers and the set of all products of the form 2^a·3^b·5^c·...

So we have ∑_n 1/n^s = ∏_p p^s/(p^s − 1).

QED