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A theorem first discovered by Leonhard Euler in the 1737 paper: "Variae observationis circa serie infinitas" ("Various observations about infinite series"). It is an important theorem that provided the bridge between number theory and analysis to produce analytic number theory, which allows one to use the techniques and methods of both classical and complex analysis to number theoretic problems.

Euler began with the Riemann zeta function (which was not called that in his day of course, as G.F.B. Riemann was still a century unborn):

             1     1    1
ζ(s) =  1 + __ +  __ + __ ....
             s     s    s
            2     3    4

He noted that writing out this infinite series involves writing out all the positive whole numbers, similar to the way Eratosthenes' Sieve begins. Multiplying both sides by (1/2s) produced:

 1        1    1    1    1
__ζ(s) = __ + __ + __ + __ + ....
 s        s    s    s    s
2        2    4    6    8

Euler then subtracted the first expression from the second, yielding:

      1              1    1    1    1
(1 - __ )ζ(s) = 1 + __ + __ + __ + __ + ...
      s              s    s    s    s
     2              3    5    7    9

All the even numbered terms, multiples of two, then disappeared from the right hand side, removing the first prime number from the list. He then repeated this process with the next unscathed term in the right hand side, 1/3s, yielding:

      1       1          1    1
(1 - __)(1 - __)ζ(s) =  __ + __ + ...
      s       s          s    s
     3       2          5    7

Now all multiples of three have been removed from the right hand side, just as in the Sieve of Eratosthenes. By repeating this process in the limit, he produced an infinite product representation of the Riemann zeta function that incorporated all of the prime numbers:

             -s  -1      -s -1      -s -1      -s -1
ζ(s) = (1 - 2   )  (1 - 3  )  (1 - 5  )  (1 - 7  )   ...

This result relates an infinite sum involving the natural numbers, the zeta function, into an infinite product running through all of the prime numbers! This result led later, mainly through the efforts of Bernhard Riemann in the mid-19th century, to propose the famed Riemann hypothesis, which related the non-trivial roots of the zeta function to the distribution of the prime numbers. It forms one of the cornerstones of modern analytic number theory, so much so that John Derbyshire calls it the "Golden Key" in his book on the Riemann Hypothesis: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.

The existence of this formula is also the basis for an elegant proof that there is no largest prime number. If the number of primes happens to be finite, the right hand side of the Euler Product Formula would be a finite product. Of course, ζ(1) is merely the harmonic series and everyone knows that series to be divergent. Now, if the left hand side is divergent, the right hand side cannot also diverge if it is a finite product, leading to a contradiction.

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