- Theorem 2
All non-trivial zeroes of
have a real part that lies in the interval
- Theorem 3
![{\displaystyle \zeta (1+it)\neq 0\forall t\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13754dbb11f84eddc2a6eea03366347f6e9a29b0)
Take the inequality,
![{\displaystyle \cos \theta \geq -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12908e0a38be15102ea63be7910c73a15e8be25c)
![{\displaystyle \implies 2(1+\cos \theta )^{2}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1bfc756715726dc9509a1a78bd7cfdd35c7535)
![{\displaystyle \implies 3+4\cos \theta +\cos 2\theta \geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faedfafecc5f7e7ae86726dbe05a5c1043900011)
Using the definition of
deduced in an earlier chapter,
![{\displaystyle \zeta (s)=\prod _{p|{\text{prime}}}{\frac {1}{1-p^{-s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0097c2ddb24a88c05797882359756a33b623e9)
Taking the log of both sides, using
![{\displaystyle \log \zeta (s)=\sum _{p|{\text{prime}}}\log \left({\frac {1}{1-p^{-s}}}\right)=-\sum _{p|{\text{prime}}}\log(1-p^{-s})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/549476d9390f591e4d9e8dc793689894f1a0e9a1)
Writing
as a power series,
![{\displaystyle \log \zeta (s)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {p^{-sn}}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1f76d6dd95def6f22805a1c1e7b36ab972ba34)
Substituting
,
![{\displaystyle \log \zeta (\sigma +it)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {p^{-n\sigma -nit}}{n}}=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\exp(-nit\log p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e5088380b5ab3c9e16b659fae8025bdd53b0e2)
Taking the modulus of the argument,
![{\displaystyle \log |\zeta (\sigma +it)|=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\Re \exp(-nit\log p)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\Re \left(\cos(nt\log p)-i\sin(nt\log p)\right)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\cos(nt\log p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55f68b2d7020e37e0b1c79e0f46f66fdcce8faf7)
Substituting appropriate values,
![{\displaystyle 3\log |\zeta (\sigma )|+4\log |\zeta (\sigma +it)|+\log |\zeta (\sigma +2it)|=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {3+4\cos(nt\log p)+\cos(2nt\log p)}{p^{n\sigma }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ac8055f090ef80d2be59632ed2e10f35c64f65)
If one lets
, it should become apparent that,
![{\displaystyle \sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {3+4\cos(nt\log p)+\cos(2nt\log p)}{p^{n\sigma }}}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d763513d604195245f302c545b76c3baaf122bb)
Clearly implying,
![{\displaystyle \log |\zeta ^{3}(\sigma )|+\log |\zeta ^{4}(\sigma +it)|+\log |\zeta (\sigma +2it)|\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/578b2f2af1199cb8235e0a228ec3dc2ae07e34c4)
Exponentiating both sides,
![{\displaystyle \zeta ^{3}(\sigma )\zeta ^{4}(\sigma +it)\zeta (\sigma +2it)\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d516faa02d2d7c7667c06cfd120baed5783a3a85)
Let's assume that
has a zero at
, then,
![{\displaystyle \lim _{\sigma \to 1^{+}}\zeta ^{3}(\sigma )\zeta ^{4}(\sigma +it_{0})\zeta (\sigma +2it_{0})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf369123ef0b0f3b10fc31daa4f40be1bf02e22b)
As
gives a pole, and
gives a zero, contradicting the previously stated inequality, proving theorem 3 by contradiction
.
- Theorem 4
![{\displaystyle \zeta (it)\neq 0\forall t\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/518ccd90364aa16cc930d83ef5cc16f330612823)
Using the functional equation,
![{\displaystyle \zeta (it)=2^{it}\pi ^{it-1}\sin \left({\frac {\pi it}{2}}\right)\Gamma (1-it)\zeta (1-it)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b226beb981af824329c9948a921226d821d42ea9)
By theorem 3, the RHS is non-zero, hence as is the LHS.
Theorems 3 and 4 are sufficient to imply theorem 2.
Riemann, knowing that all zeroes lied in the critical strip, postulated,
- Conjecture
All non-trivial zeroes of
have a real part of
The above conjecture is considered to be the most notable in pure mathematics, and the most notable of Riemann's works.