- Theorem 1
![{\displaystyle \zeta (s)=\prod _{p|{\text{prime}}}{\frac {1}{1-p^{-s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0097c2ddb24a88c05797882359756a33b623e9)
Where
denotes a product over the primes.
- Proof
Note that,
![{\displaystyle {\frac {1}{2^{s}}}\zeta (s)={\frac {1}{2^{s}}}+{\frac {1}{4^{s}}}+{\frac {1}{6^{s}}}\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e4f99ae1c80eeb8152f5e02417f809c4682f6f)
And hence,
![{\displaystyle \left(1-{\frac {1}{2^{s}}}\right)\zeta (s)=1+{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64029235e01c60a04f52243198d0fd52c9b2423)
One will notice that every other term has been removed. It then follows that,
![{\displaystyle {\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}}}\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6b4bbc48c8c49dff26a961df62e1a41bf1706d)
Subtracting,
![{\displaystyle \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}}}\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8647e012d99552794f24e414453b64bb86ebf9f9)
One may notice that this process sieves the RHS, meaning that,
![{\displaystyle \left(\prod _{p|{\text{prime}}}1-p^{-s}\right)\zeta (s)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c0050599b9610281451866a0d9e4f1ca695eeb)
It therefore follows that,
![{\displaystyle \blacksquare }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8733090f2d787d03101c3e16dc3f6404f0e7dd4c)