The so-called “fundamental theorem of arithmetic” states than any integer is a product of powers of prime numbers in a unique way. Euler was (apparently) the first to realize that this fact could be expressed as an identity between an infinite sum and an infinite product. Specifically, Euler discovered:
∑1≤n<∞ 1/ns = ∏p (1 – 1/ps)-1
This formula is valid only for s with Re(s) > 1, so that both sides “converge” to finite values.
If you’ve read this far, you’ve probably had enough mathematics to be comfortable with the ∑-notation, but may not be familiar with the product notation on the right side of this equation. It should be intuitively clear, however, that it may be understood in an analogous way as a product of terms of the indicated form. In this case, the product is taken over all positive primes, and only primes. (Note that 1 is not considered a prime, so the product makes sense.) For an infinite product, there are major issues of convergence, just as for infinite sums. Euler worked mostly formally and didn’t bother much about such issues. Nevertheless, the issues can be dealt with and a rigorous treatment of infinte products can be given, just as for infinite sums.
To “prove” Euler’s equation, in a loose formal sense, we need only recall the so-called “geometric series”:
1/(1-x) = ∑0≤n<∞ xn = 1 + x + x2 + …
The “proof” of this latter identity is obtained by multiplying both sides by 1 – x and rearranging terms, so that
1 = (1 – x)(1 + x + x2 + …) = (1 + x + x2 + …) – (x + x2 + x3 + …)
On the right hand side of this, all terms cancel out except for the 1. These formal operations can be rigorously justified if x is a real (or complex) number of absolute value less than 1 (i. e. |x| < 1). If p is any prime, 1/p < 1, and in fact (less obviously) |1/ps| < 1. So putting x = 1/ps in the geometric series gives
(1 – 1/ps)-1 = ∑0≤k<∞ (1/ps)k = ∑0≤k<∞ p-ks
Finally, taking the product of all these terms gives
∏p (1 – 1/ps)-1 = ∏p ∑0≤k<∞ p-ks = (1 + 2-s + 2-2s + …) × (1 + 3-s + 3-2s + …) × …
Now, when you multiply everything out on the right side of that, you find that a term of the form 1/ns occurs exactly once for each n such that 1 ≤ n < ∞, because each such integer is uniquely expressible as a product of powers of primes. (You might have to stare at that and scratch your head for awhile to see it.) This proves Euler’s product formula.
The existence of this product formula of Euler to represent the zeta function is the simplest, most obvious way in which the zeta function is connected with prime numbers. It is certainly not the only way, nor – by a long shot – the most profound. To begin to appreciate deeper connections, we have to look into what is known about the distribution of prime numbers.