  In 1737 Leonard Euler gave what we often now think of as a new proof, based
on infinite series, of Euclid's theorem that there are infinitely many prime
numbers. Our short paper uses a simple modification of Euler's argument to
obtain new results about the distribution of prime factors of sets of integers,
including a weak one-sided Tschebyshev inequality. An example shows that there
cannot be a prime number theorem in this situation, or even a pair of
Tschebyshev inequalities, but it would be very interesting to know if a
one-sided Tschebyshev inequality holds.
