New Results on Primes from an Old Proof of Euler's

Abstract

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.