Editing Riemann zeta function (section)

==Euler's product formula==
In 1737, the connection between the zeta function and [[prime number]]s was discovered by Euler, who [[Proof of the Euler product formula for the Riemann zeta function|proved the identity]]

:<math>\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}},</math>

where, by definition, the left hand side is {{math|''ζ''(''s'')}} and the [[infinite product]] on the right hand side extends over all prime numbers {{mvar|p}} (such expressions are called [[Euler product]]s):

:<math>\prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = \frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}} \cdots \frac{1}{1-p^{-s}} \cdots</math>

Both sides of the Euler product formula converge for {{math|Re(''s'') > 1}}. The [[Proof of the Euler product formula for the Riemann zeta function|proof of Euler's identity]] uses only the formula for the [[geometric series]] and the [[fundamental theorem of arithmetic]]. Since the [[harmonic series (mathematics)|harmonic series]], obtained when {{math|''s'' {{=}} 1}}, diverges, Euler's formula (which becomes {{math|Π<sub>''p''</sub> {{sfrac|''p''|''p'' − 1}}}}) implies that there are [[Euclid's theorem|infinitely many primes]].<ref>{{cite book|first=Charles Edward |last=Sandifer |title=How Euler Did It |publisher=Mathematical Association of America |date=2007 |page=193 |isbn=978-0-88385-563-8}}</ref> Since the logarithm of {{math|{{sfrac|''p''|''p'' − 1}}}} is approximately {{math|{{sfrac|1|''p''}}}}, 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 Eratosthenes]] 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 density|asymptotic probability]] that {{mvar|s}} randomly selected integers are set-wise [[coprime]]. Intuitively, the probability that any single number is divisible by a prime (or any integer) {{mvar|p}} is {{math|{{sfrac|1|''p''}}}}. Hence the probability that {{mvar|s}} numbers are all divisible by this prime is {{math|{{sfrac|1|''p''{{isup|''s''}}}}}}, and the probability that at least one of them is ''not'' is {{math|1 − {{sfrac|1|''p''{{isup|''s''}}}}}}. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors {{mvar|n}} and {{mvar|m}} [[if and only if]] it is divisible by&nbsp;{{mvar|nm}}, an event which occurs with probability&nbsp;{{math|{{sfrac|1|''nm''}}}}). Thus the asymptotic probability that {{mvar|s}} numbers are coprime is given by a product over all primes,

: <math>\prod_{p \text{ prime}} \left(1-\frac{1}{p^s}\right) = \left( \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \right)^{-1} = \frac{1}{\zeta(s)}. </math>