Take the 2-minute tour ×  In laymen's terms, as much as
possible: What is the Riemann-Zeta function, and why does it
come up so often with relation to prime numbers?  share|improve
this question A good example of a question that is asked by
someone genuinely interested in math but is looking for an
accessible way into more advanced number theory that he would
otherwise have no other means of finding. –  Justin L. Jul 23
'10 at 6:47 add comment  5 Answers  up vote 49 down vote
accepted  Suppose you want to put a probability distribution on
the natural numbers for the purpose of doing number theory. What
properties might you want such a distribution to have? Well, if
you're doing number theory then you want to think of the prime
numbers as acting "independently": knowing that a number is
divisible by $p$ should give you no information about whether
it's divisible by $q$.  That quickly leads you to the following
realization: you should choose the exponent of each prime in the
prime factorization independently. S