Apéry's theorem is the discovery that the number

is an irrational real number. Here z is the Riemann zeta-function. This important function is defined in the right half plane {s = + it ∈ C: 1 < } by the convergent series

and by the convergent infinite product

where the product is over the set of all prime numbers p. The function s z(s) is an analytic function of the complex variable s = + it in the half plane 1 < . Because the zeta-function is defined by both the infinite series and the infinite product, it provides a means for using techniques from complex analysis in order to investigate questions about prime numbers.

There are many elementary problems of interest about the zeta-function that have no immediate implications for the distribution of primes. One of these is to determine the values of z(s) at the integers...