Theorem 1

${\displaystyle \zeta (-2s)=0\forall s\in \mathbb {N} }$

Proof

Consider the functional equation for Zeta,

${\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)}$

Notice that for ${\displaystyle \zeta (-2s)}$, the sine term evaluates to ${\displaystyle \sin(-\pi s)}$ which evaluates to 0 for all integers ${\displaystyle s}$, hence ${\displaystyle \zeta (-2s)=0}$ for all natural ${\displaystyle s}$ ${\displaystyle \blacksquare }$.

Definition 1

These zeroes are referred to as trivial zeroes. As a set,

${\displaystyle \zeta _{t}=\{-2s:s\in \mathbb {N} \}}$

Zeroes that do not lie in this set are to be referred to as non-trivial zeroes.

Note 1

The above argument cannot be applied to ${\displaystyle (s<1)\in \mathbb {Z} }$, as ${\displaystyle \zeta (1)}$ is a simple pole (${\displaystyle s=0}$), as are negative arguments of ${\displaystyle \Gamma }$ (${\displaystyle s<0}$).

Theorem 2

All non-trivial zeroes of ${\displaystyle \zeta }$ have a real part that lies in the interval ${\displaystyle (0,1)}$

Theorem 3

${\displaystyle \zeta (1+it)\neq 0\forall t\in \mathbb {R} }$

Take the inequality,

${\displaystyle \cos \theta \geq -1}$

${\displaystyle \implies 2(1+\cos \theta )^{2}\geq 0}$

${\displaystyle \implies 3+4\cos \theta +\cos 2\theta \geq 0}$

Using the definition of ${\displaystyle \zeta }$ deduced in an earlier chapter,

${\displaystyle \zeta (s)=\prod _{p|{\text{prime}}}{\frac {1}{1-p^{-s}}}}$

Taking the log of both sides, using ${\displaystyle \log \left(\prod f(x)\right)=\sum \log \left(f(x)\right)}$

${\displaystyle \log \zeta (s)=\sum _{p|{\text{prime}}}\log \left({\frac {1}{1-p^{-s}}}\right)=-\sum _{p|{\text{prime}}}\log(1-p^{-s})}$

Writing ${\displaystyle \log }$ as a power series,

${\displaystyle \log \zeta (s)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {p^{-sn}}{n}}}$

Substituting ${\displaystyle s=\sigma +it}$,

${\displaystyle \log \zeta (\sigma +it)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {p^{-n\sigma -nit}}{n}}=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\exp(-nit\log p)}$

Taking the modulus of the argument,

${\displaystyle \log |\zeta (\sigma +it)|=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\Re \exp(-nit\log p)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\Re \left(\cos(nt\log p)-i\sin(nt\log p)\right)=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {1}{p^{n\sigma }}}\cos(nt\log p)}$

Substituting appropriate values,

${\displaystyle 3\log |\zeta (\sigma )|+4\log |\zeta (\sigma +it)|+\log |\zeta (\sigma +2it)|=\sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {3+4\cos(nt\log p)+\cos(2nt\log p)}{p^{n\sigma }}}}$

If one lets ${\displaystyle nt\log p=\theta }$, it should become apparent that,

${\displaystyle \sum _{p|{\text{prime}}}\sum _{n=1}^{\infty }{\frac {3+4\cos(nt\log p)+\cos(2nt\log p)}{p^{n\sigma }}}\geq 0}$

Clearly implying,

${\displaystyle \log |\zeta ^{3}(\sigma )|+\log |\zeta ^{4}(\sigma +it)|+\log |\zeta (\sigma +2it)|\geq 0}$

Exponentiating both sides,

${\displaystyle \zeta ^{3}(\sigma )\zeta ^{4}(\sigma +it)\zeta (\sigma +2it)\geq 1}$

Let's assume that ${\displaystyle \zeta }$ has a zero at ${\displaystyle 1+it_{0}}$, then,

${\displaystyle \lim _{\sigma \to 1^{+}}\zeta ^{3}(\sigma )\zeta ^{4}(\sigma +it_{0})\zeta (\sigma +2it_{0})=0}$

As ${\displaystyle \lim _{\sigma \to 1^{+}}\zeta (\sigma )}$ gives a pole, and ${\displaystyle \zeta (1+it_{0})}$ gives a zero, contradicting the previously stated inequality, proving theorem 3 by contradiction ${\displaystyle \blacksquare }$.

Theorem 4

${\displaystyle \zeta (it)\neq 0\forall t\in \mathbb {R} }$

Using the functional equation,

${\displaystyle \zeta (it)=2^{it}\pi ^{it-1}\sin \left({\frac {\pi it}{2}}\right)\Gamma (1-it)\zeta (1-it)}$

By theorem 3, the RHS is non-zero, hence as is the LHS. ${\displaystyle \blacksquare }$

Theorems 3 and 4 are sufficient to imply theorem 2. ${\displaystyle \blacksquare }$

Conjecture

All non-trivial zeroes of ${\displaystyle \zeta }$ have a real part of ${\displaystyle {\frac {1}{2}}}$