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[[Leonhard Euler]] proved the '''[[Riemann_zeta_function#Euler_product_formula|Euler product formula for the Riemann zeta function]]''' in his thesis ''Variae observationes circa series infinitas'' (''Various Observations about Infinite Series''), published by St Petersburg Academy in 1737.<ref>{{cite web | title = A history of calculus |author=O'Connor, J.J. and Robertson, E.F. | publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |date=February 1996|accessdate= 2007-08-07}}</ref><ref>[[John Derbyshire]] (2003), chapter 7, "The Golden Key, and an Improved Prime Number Theorem"</ref>

==The Euler product formula==

The Euler product formula for the [[Riemann zeta function]] reads

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

where the left hand side equals the Riemann zeta function:

:<math>\zeta(s) = \sum_{n=1}^\infty\frac{1}{n^s} = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+ \ldots</math>
and the product on the right hand side extends over all [[prime number]]s ''p'':

:<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}} \cdots \frac{1}{1-p^{-s}} \cdots</math>

== Proof of the Euler product formula ==
[[File:Sieve_of_Eratosthenes_animation.gif|frame|right|The method of [[Eratosthenes]] used to sieve out prime numbers is employed in this proof.]]
This sketch of a [[mathematical proof|proof]] only makes use of simple algebra commonly taught in high school. This was originally the method by which [[Leonhard Euler|Euler]] discovered the formula. There is a certain [[sieve of Eratosthenes|sieving]] property that we can use to our advantage:

:<math>\zeta(s) = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+ \ldots </math>

:<math>\frac{1}{2^s}\zeta(s) = \frac{1}{2^s}+\frac{1}{4^s}+\frac{1}{6^s}+\frac{1}{8^s}+\frac{1}{10^s}+ \ldots </math>

Subtracting the second from the first we remove all elements that have a factor of 2:

:<math>\left(1-\frac{1}{2^s}\right)\zeta(s) = 1+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{9^s}+\frac{1}{11^s}+\frac{1}{13^s}+ \ldots </math>

Repeating for the next term:

:<math>\frac{1}{3^s}\left(1-\frac{1}{2^s}\right)\zeta(s) = \frac{1}{3^s}+\frac{1}{9^s}+\frac{1}{15^s}+\frac{1}{21^s}+\frac{1}{27^s}+\frac{1}{33^s}+ \ldots </math>

Subtracting again we get:

:<math>\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{2^s}\right)\zeta(s) = 1+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{11^s}+\frac{1}{13^s}+\frac{1}{17^s}+ \ldots </math>

where all elements having a factor of 3 or 2 (or both) are removed.

It can be seen that the right side is being sieved. Repeating infinitely we get:

:<math> \ldots \left(1-\frac{1}{11^s}\right)\left(1-\frac{1}{7^s}\right)\left(1-\frac{1}{5^s}\right)\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{2^s}\right)\zeta(s) = 1 </math>

Dividing both sides by everything but the ζ(''s'') we obtain:

:<math> \zeta(s) = \frac{1}{\left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{5^s}\right)\left(1-\frac{1}{7^s}\right)\left(1-\frac{1}{11^s}\right) \ldots } </math>

This can be written more concisely as an infinite product over all primes ''p'':
:<math>\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}</math>

To make this proof rigorous, we need only observe that when <math>\Re(s) > 1</math>, the sieved right-hand side approaches 1, which follows immediately from the convergence of the [[Dirichlet series]] for ζ(''z'').

== The case <math>s = 1 </math>==

An interesting result can be found for ζ(1)

:<math> \ldots \left(1-\frac{1}{11}\right)\left(1-\frac{1}{7}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{2}\right)\zeta(1) = 1 </math>
which can also be written as,
:<math> \ldots \left(\frac{10}{11}\right)\left(\frac{6}{7}\right)\left(\frac{4}{5}\right)\left(\frac{2}{3}\right)\left(\frac{1}{2}\right)\zeta(1) = 1 </math>
which is,
:<math> \left(\frac{\ldots\cdot10\cdot6\cdot4\cdot2\cdot1}{\ldots\cdot11\cdot7\cdot5\cdot3\cdot2}\right)\zeta(1) = 1 </math>
as,
<math>\zeta(1) = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+ \ldots </math>

thus,

:<math> 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+ \ldots = \frac{2\cdot 3\cdot 5\cdot 7\cdot 11\cdot\ldots}{1\cdot 2\cdot 4\cdot 6\cdot 10\cdot\ldots} </math>

We know that the left-hand side of the equation diverges to infinity, therefore the numerator on the right-hand side (the [[primorial]]) must also be infinite for divergence. This proves that there are [[infinitude of primes|infinitely many prime numbers]].

== Another proof ==

Each factor (for a given prime ''p'') in the product above can be expanded to a [[geometric series]] consisting of the reciprocal of ''p'' raised to multiples of ''s'', as follows

:<math>\frac{1}{1-p^{-s}} = 1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \frac{1}{p^{3s}} + \ldots + \frac{1}{p^{ks}} + \ldots </math>

When <math>\Re(s) > 1</math>, we have |''p''<sup>−''s''</sup>| < 1 and this series [[series (mathematics)|converges absolutely]]. Hence we may take a finite number of factors, multiply them together, and rearrange terms. Taking all the primes ''p'' up to some prime number limit ''q'', we have

:<math>\left|\zeta(s) - \prod_{p \le q}\left(\frac{1}{1-p^{-s}}\right)\right| < \sum_{n=q+1}^\infty \frac{1}{n^\sigma}</math>

where σ is the real part of ''s''. By the [[fundamental theorem of arithmetic]], the partial product when expanded out gives a sum consisting of those terms ''n''<sup>−''s''</sup> where ''n'' is a product of primes less than or equal to ''q''. The inequality results from the fact that therefore only integers larger than ''q'' can fail to appear in this expanded out partial product. Since the difference between the partial product and ζ(''s'') goes to zero when σ > 1, we have convergence in this region.

==References==
* [[John Derbyshire]], ''[[Prime Obsession|Prime Obsession: Bernhard Riemann and The Greatest Unsolved Problem in Mathematics]]'', Joseph Henry Press, 2003, ISBN 978-0-309-08549-6

==Notes==
<references/>

[[Category:Zeta and L-functions]]
[[Category:Article proofs]]

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