Riemann Zeta Has No Zeroes With Real Part One

Theorem
Let $\zeta$ be the Riemann zeta function.

Then for all $t \in \R$:
 * $\map \zeta {1 + i t} \ne 0$

Proof
Throughout, the complex variable $s$ is $s = \sigma + i t$.

We have, for $\sigma > 1$,

Therefore,


 * $(1): \quad \ds -\map \Re {\dfrac {\map {\zeta'} s} {\map \zeta s} } = \sum_{n \mathop \ge 1} \map \Lambda n n^{-\sigma} \map \cos {t \log n}$

Now observe that:


 * $3 + 4 \cos \theta + \map \cos {2 \theta} = 2 \paren {1 + \cos \theta}^2 \ge 0$

Because for all $n \ge 1$ we have $\map \Lambda n n^{-\sigma} \ge 0$, we have:

Now let:


 * $\map \eta s = \map \zeta s^3 \cdot \map \zeta {s + i t}^4\cdot \map \zeta {s + 2 i t}$

Then the above computation has shown that


 * $\map \Re {\dfrac {\map {\eta'} s} {\map \eta s} } \le 0$

By Poles of Riemann Zeta Function we know that $\zeta$ has a simple pole at $s = 1$ with residue $1$.

that $1 + i t$ is a zero of $\zeta$ of order $d \ge 1$.

Therefore, at $s = 1$, $\eta$ has a zero of order $4 d - 3 \ge 0$, that is:


 * $\map \eta s \sim (s - 1)^{4 d - 3}$

as $s \to 1^+$, where:
 * $\sim$ indicates asymptotic equality
 * superscript $+$ denotes a limit from the right along the real line.

Therefore:


 * $\dfrac {\map {\eta'} s} {\map \eta s} \sim \dfrac {4 d - 3} {s - 1}$

as $s \to 1^+$.

Since $\map \Re {\dfrac {4 d -3} {s - 1} } \to + \infty$ as $s \to 1^+$, it follows that:


 * $\map \Re {\dfrac {\map {\eta'} s} {\map \eta s} } \to \infty$

as $s \to 1^+$.

But we have already shown that:
 * $\map \Re {\dfrac {\map {\eta'} s} {\map \eta s} } \le 0$

which is a contradiction.