Proof of the Euler product formula for the Riemann zeta function

We will prove that the following formula holds::

where ζ denotes the Riemann zeta function and the product extends over all prime numbers "p".

[
Eratosthenes used to sieve out prime numbers is employed in this proof.]

Proof of the Euler product formula

This sketch of a proof only makes use of simple algebra that most high school students can understand. This was originally the method by which Euler discovered the formula. There is a certain sieving property that we can use to our advantage:

:$zeta(s)\; =\; 1+frac\{1\}\{2^s\}+frac\{1\}\{3^s\}+frac\{1\}\{4^s\}+frac\{1\}\{5^s\}+\; cdots$

:$frac\{1\}\{2^s\}zeta(s)\; =\; frac\{1\}\{2^s\}+frac\{1\}\{4^s\}+frac\{1\}\{6^s\}+frac\{1\}\{8^s\}+frac\{1\}\{10^s\}+\; cdots$

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

:$left(1-frac\{1\}\{2^s\}\; ight)zeta(s)\; =\; 1+frac\{1\}\{3^s\}+frac\{1\}\{5^s\}+frac\{1\}\{7^s\}+frac\{1\}\{9^s\}+frac\{1\}\{11^s\}+frac\{1\}\{13^s\}+\; cdots$

Repeating for the next term:

:$frac\{1\}\{3^s\}left(1-frac\{1\}\{2^s\}\; ight)zeta(s)\; =\; frac\{1\}\{3^s\}+frac\{1\}\{9^s\}+frac\{1\}\{15^s\}+frac\{1\}\{21^s\}+frac\{1\}\{27^s\}+frac\{1\}\{33^s\}+\; cdots$

Subtracting again we get:

:$left(1-frac\{1\}\{3^s\}\; ight)left(1-frac\{1\}\{2^s\}\; ight)zeta(s)\; =\; 1+frac\{1\}\{5^s\}+frac\{1\}\{7^s\}+frac\{1\}\{11^s\}+frac\{1\}\{13^s\}+frac\{1\}\{17^s\}+\; cdots$

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:

:$cdots\; left(1-frac\{1\}\{11^s\}\; ight)left(1-frac\{1\}\{7^s\}\; ight)left(1-frac\{1\}\{5^s\}\; ight)left(1-frac\{1\}\{3^s\}\; ight)left(1-frac\{1\}\{2^s\}\; ight)zeta(s)\; =\; 1$

Dividing both sides by everything but the $zeta(s)$ we obtain:

:$zeta(s)\; =\; frac\{1\}\{left(1-frac\{1\}\{2^s\}\; ight)left(1-frac\{1\}\{3^s\}\; ight)left(1-frac\{1\}\{5^s\}\; ight)left(1-frac\{1\}\{7^s\}\; ight)left(1-frac\{1\}\{11^s\}\; ight)\; cdots\; \}$

This can be written more concisely as an infinite product over all primes "p":

:$zeta(s);=;prod\_\{p\}\; (1-p^\{-s\})^\{-1\}.$

To make this proof rigorous, we need only observe that when Re(s) > 1, the sieved right-hand side approaches 1, which follows immediately from the convergence of the Dirichlet series for $zeta(s)$.

An interesting result can be found for $zeta(1)$

:$cdots\; left(1-frac\{1\}\{11\}\; ight)left(1-frac\{1\}\{7\}\; ight)left(1-frac\{1\}\{5\}\; ight)left(1-frac\{1\}\{3\}\; ight)left(1-frac\{1\}\{2\}\; ight)zeta(1)\; =\; 1$which can also be written as,:$cdots\; left(frac\{10\}\{11\}\; ight)left(frac\{6\}\{7\}\; ight)left(frac\{4\}\{5\}\; ight)left(frac\{2\}\{3\}\; ight)left(frac\{1\}\{2\}\; ight)zeta(1)\; =\; 1$which is, :$left(frac\{...\; 10.6.4.2.1\}\{...\; 11.7.5.3.2\}\; ight)zeta(1)\; =\; 1$as,$zeta(1)\; =\; 1+frac\{1\}\{2\}+frac\{1\}\{3\}+frac\{1\}\{4\}+frac\{1\}\{5\}+\; cdots$

thus,

:$1+frac\{1\}\{2\}+frac\{1\}\{3\}+frac\{1\}\{4\}+frac\{1\}\{5\}+\; cdots\; ;=;\; left(frac\{...\; 11.7.5.3.2\}\{...\; 10.6.4.2.1\}\; ight)$

We know that the left-hand side of the equation diverges to infinity therefore the numerator on the right-hand side (the series of primes) must also be infinite for divergence.

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

:$frac\{1\}\{1-p^\{-s\; =\; 1\; +\; frac\{1\}\{p^s\}\; +\; frac\{1\}\{p^\{2s\; +\; frac\{1\}\{p^\{3s\; +\; cdots\; +\; frac\{1\}\{p^\{ks\; +\; cdots$

When $zeta(s)\; >\; 1$, and this series 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

:

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 $frac\{1\}\{n^s\}$ 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.