What is the sum of the reciprocals of all prime numbers? (i.e. 1/2 + 1/3 + 1/5 + 1/7...)

Even without rigorous proof it would seem that this series diverges. The informal reasoning is that the density of prime numbers in the vicinity of x is 1/ln(x). If we multiply this density function by the function 1/x, representing the values in the series itself, we get 1/(x ln x) to be integrated out to infinity. But ∫(1/(x ln x)) dx = ln(ln(x)), and this function increases without limit as x grows larger, albeit very slowly.

Posted by Charlie on 2003-12-24 15:54:34