.
If the two numbers are coprime, then for each prime p, p does not divide both
numbers. If we assume that being divisible by one prime is independent of
being divisible by another, then the probability of the numbers being coprime
is
.
Let us define a generalization of the reciprocal of this as the Riemann zeta
function:
where the product is over all primes less than our numbers. Assuming the numbers
are large, and letting the product be over all primes, we can define this
as the zeta function. It turns out that the zeta function can also be written
as a sum:
This relationship can be proved by expanding each term in the product as a power series. When the power series are multiplied together, we use the fundamental theorem of arithmetic, that each integer is expressible as a product of powers of primes, and each product of powers of primes represnts just one integer. This product formula for the zeta function provides a bridge from the properties of certain integers (prime numbers), to a function that depends on a real number (actually a complex number), which may be examined with the powerful tools of calculus and analytic function theory.
The value of the zeta function at s=2 is then the reciprocal of the probability
that we seek.
Taking a function like this:
and expanding it as a Fourier series can provide a value for zeta(2).
The functionis piecewise quadratic but also periodic, and the value of the
Fourier series at the cusp shows that zeta(2) is 
, so that
the reciprocal of this is the probability of two large numbers being coprime.