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

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

2002-10-18

arxiv.org/abs/math/0210282v2

Mathematics

Number Theory

Revision 1, Plain Tex, 7 pages. Revision 1 corrects history, references and non-computability example

Scientific paper

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.

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