Integers Relatively Prime to the First n Primes
The probability that a prime number does not divide a natural number is . Hence, the probability that a natural number is relatively prime to all primes less than equals . Legendre proved that the large limit of this product is zero, meaning that the probability that a large random integer is a prime approaches zero. This Demonstration computes the probability that an integer is coprime with each of the first primes.
Riemann's zeta function is defined by . Legendre's theorem states that .