where
is the th
prime. This is Euler's product (Whittaker and Watson
1990), called by Havil (2003, p. 61) the "all-important formula" and
by Derbyshire (2004, pp. 104-106) the "golden key."
This can be proved by expanding the product, writing each term as a geometric
series, expanding, multiplying, and rearranging terms,
which can be seen by expanding the product to obtain
(5)
(6)
(7)
(8)
(9)
, but the finite product
exists, giving
(10)
For upper limits ,
1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072,
... (OEIS A060753 and A038110).
Premultiplying by
and letting
gives a beautiful result known as the Mertens theorem.
The Euler product appears briefly in a pan of John Nash's (played by Russell Crowe) blackboard scribblings in Ron Howard's 2001 film A
Beautiful Mind.