Euler Product Formula
The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity
where, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):
Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler's formula (which becomes ) implies that there are infinitely many primes.
The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer), p is 1/p. Hence the probability that s numbers are all divisible by this prime is 1/ps, and the probability that at least one of them is not is 1 − 1/ps. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability 1/(nm).) Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,
(More work is required to derive this result formally.)
Read more about this topic: Riemann Zeta Function
Famous quotes containing the words product and/or formula:
“A gentleman opposed to their enfranchisement once said to me, “Women have never produced anything of any value to the world.” I told him the chief product of the women had been the men, and left it to him to decide whether the product was of any value.”
—Anna Howard Shaw (1847–1919)
“The formula for achieving a successful relationship is simple: you should treat all disasters as if they were trivialities but never treat a triviality as if it were a disaster.”
—Quentin Crisp (b. 1908)