Euler discovered there is an alternative formula or recipe for the zeta function which depends on knowing the prime numbers:
(1) calculate p^x for each prime number p
(2) take the reciprocal of all the numbers in (1)
(3) take 1 away from all the numbers in (2)
(4) take the reciprocal of all the numbers in step (3)
(5) multiply all the numbers in step (4) together

That the two recipes cook the same cake depends on the fact that all the numbers 1,2,3,4… can be written as prime numbers multiplied together.

For example the number 60 is 2x2x3x5.

Euler realised that every term in his infinite sum like 1/60 could be pulled apart using the property that 60 is built out of its prime building blocks. So he wrote:

1/60=(1/2)x(1/2)x(1/3)x(1/5)
= (1/2²)x(1/3)x(1/5)

But if he did this to every term in his infinite sum he could pull all the calculations apart and write them as:

1+1/2+1/3+…+1/60+…=
(1+1/2+1/2²+…)(1+1/3+1/3²+…)(1+1/5+1/5²+…)…(1+1/p+1/p²+…)…

where each bracket contains all the powers of a particular prime. So each fraction 1/N in the infinite sum is built by multiplying a fraction from each bracket. If p is one of the building blocks of the number N appearing A times then take the fraction 1/p^A from the p-bracket. If p is not a building block then simply choose 1.

You can try this on the first 10 fractions of the harmonic series:

1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+…
=(1+1/2+1/4+1/8+…)(1+1/3+1/9+…)(1+1/5+…)(1+1/7+…)(1+1/11+…

To get one of the fractions in the harmonic series, you have to pick one term from each bracket and multiply them together to get your choice from the harmonic series. For example to get 1/10, you'll choose 1/2 from the 2-bracket and 1/5 from the 5-bracket and 1 from every other bracket.

OK, that admittedly was quite a heavy burst of equations. What you should take away from this is Euler's observation that the infinite sums in which he was interested have connections with the primes because by multiplying together all the primes in different combinations you can reconstruct Euler's sums. It is somehow a very neat way to express in one infinite equation the Greeks' observation that numbers are built out of primes. Euler's expression, known today as Euler's product, can be summed up as "Adding fractions=Multiplying reciprocals of primes".

It was when he took the logarithm of the harmonic series that Euler discovered this cryptic expression for the sum of the reciprocals of primes that began our tour of these infinite sums. Since the harmonic sum was log, taking the logarithm again gave log log. But on the other side of his formula for the harmonic series he had expressions multiplied together depending on the primes. Taking the logarithm changed this multiplication into addition and, hey presto, he got the sum of the reciprocals of primes. So adding up the reciprocals of the first N primes Euler believed would get him as far as log log(N). So although Euler knew that the sum eventually gets to infinity, even if we sum the reciprocals of all primes known to date, we only get as far as about 4.

His argument for why the sum of the reciprocals of primes is log(log ) used the infinite product representation for the harmonic series in terms of primes:

1+1/2+1/3+1/4+…=
(1+1/2+1/4+1/8+…)(1+1/3+1/9+…)…(1+1/p+1/p²+…)…

Euler took the logarithm of both sides. Since the harmonic series on the left is log, taking its logarithm gives log log. On the right hand side, Euler used the special fact about logarithms which made them so powerful for engineers: log(AxB)=logA+logB. He also knew that log(1+1/p+1/p²+…) is approximately 1/p. So the logarithm of the right hand side is essentially the sum of the reciprocals of the primes. Hence
1/2+1/3+1/5+1/7+1/11+…=log(log ).

You may say that Euler's expression for the harmonic series in terms of primes has not got us much further. It is after all just another way of expressing something the Greeks knew two thousand years previously. But by turning this Greek gem and staring at it from a 18th Century angle a new perspective emerged that the Greek's would never have guessed at. Indeed Euler too was not to grasp the full significance of his rewriting of this property of the primes. This took another 100 years and the insight of Bernhard Riemann.