The first inklings of a connection between primes and logarithms actually appear even before Gauss and Legendre. In his Introductio in analysin infinitorum published in 1748, Euler recorded a rather cryptic sum:

1/2+1/3+1/5+1/7+1/11+…=log(log).

He divided 1 by each of the primes, called the reciprocal of a prime, and then added up all these numbers. Euler claimed that the total sum is the logarithm of the logarithm of infinity. This strange number is still infinity but Euler tried to capture how fast this sum climbs to infinity. His claim was that by adding up all the reciprocals of primes below N the sum will be about the same size as taking the logarithm of the logarithm of N. So not very fast but eventually this sum Euler claims gets as big as you want creeping very slowly to infinity.

If the Greeks had proved there are infinitely many primes, you might think that it is obvious that adding together the infinite list of reciprocals of primes you would get an infinite answer. But it isn't necessarily the case that adding together an infinite number of things you will be able to make the sum as big as you want.

There is an example of such a sequence of numbers which has its origins in another Greek story, that of the race between the tortoise and Achilles. The Greek philosopher Zeno contended that if the tortoise has a head start, Achilles would never be able to overtake the tortoise. His argument was that by the time Achilles had reached the tortoise's starting place, the tortoise had moved on to a second point ahead of Achilles. But by the time Achilles had reached this second spot, the tortoise would still be ahead having reached a third point. Zeno's paradox relies on convincing you that an infinite number of actions must take an infinite amount of time to complete so he can't overtake the tortoise.

But the reality is of course that Achilles overtakes the tortoise with no trouble. The answer lies in the fact that if we add up the time it takes to complete the infinite number of actions that Zeno demands of Achilles, the total time is not infinite. So Achilles can complete his task of overtaking the tortoise in a finite amount of time.

Suppose the tortoise is travelling at half the speed of Achilles (rather a speedy tortoise admittedly). The tortoise gets a head start. Suppose that it takes Achilles 1 second to reach the tortoise's starting point. The tortoise, travelling at half the speed has got to a second point. It takes Achilles just 1/2 second to reach this second point. Continuing like this, the race is broken up into an infinite number of stages, each of which takes Achilles half as long to complete as the previous stage. To overtake the tortoise will therefore take Achilles
1+1/2+1/4+1/8+1/16+…+1/2ⁿ +…
seconds.

Zeno's paradox is resolved because this infinite sum adds up to 2 seconds. To see this, take the sum of the first n terms and call this a_n. Then we do something which at first sight looks completely useless. Write

a_n =2 a_n - a_n.

But what is 2 times a_n? It is 2x(1+1/2+1/4+…+1/2ⁿ)=2+1+1/2+…+1/2^n-1, i.e. multiplying by 2 somehow shifts all the terms along one. But now look what happens when we subtract the expression for a_n: most of the terms magically disappear!

a_n =2 a_n - a_n
=2+1+1/2+…+1/2^n-1
-(1+1/2+…+1/2ⁿ)
=2-1/2ⁿ

Hence a_n is smaller than 2 but as n gets bigger it gets closer to 2 since 1/2ⁿ gets very small. This sort of argument is very typical of the power of just turning an object to look at it from a slightly different angle from which everything becomes transparent. Who would have thought there would be much mileage to be had from writing the number a_n as a_n + a_n - a_n =2 a_n - a_n. It is this lateral thinking that one finds in puzzle solvers that can translate into a powerful ability to navigate the mathematical world.