with residue . Euler's product formula In 1737, 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 and the infinite product on the right hand side extends over all prime numbers (such expressions are called Euler products): Both sides of the Euler product formula converge for . 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 , diverges, Euler's formula (which becomes ) implies that there are infinitely many primes. Since the logarithm of is approximately , the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the sieve of Erasthosthenes shows that the density of the set of primes within the set of positive integers is zero. The Euler product formula can be used to calculate the asymptotic probability that randomly selected integers are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer) is . Hence the probability that numbers are all divisible by this prime is , and the probability that at least one of them is not is . Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors and if and only if it is divisible by , an event which occurs with probability ). Thus the asymptotic probability that numbers are coprime is given by a product over all primes, Riemann's functional equation This zeta function satisfies the functional equation where is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points and , in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that has a simple zero at each even negative integer , known as the trivial zeros of . When is an even positive integer, the product on the right is non-zero because has a simple pole, which cancels the simple zero of the sine factor. The functional equation was established by Riemann in his 1859 paper "On the Number