Riemann Zeta Has No Zeroes With Real Part One

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

Then for all $t \in \R$, $\zeta(1 + it) \neq 0$

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

We have, for $\sigma > 1$,

Therefore,


 * $\displaystyle -\Re\left( \frac{\zeta'(s)}{\zeta(s)} \right) = \sum_{n\geq 1} \Lambda(n) n^{-\sigma} \cos(t\log n) \qquad (1)$

Now observe that


 * $3 + 4\cos\theta + \cos(2\theta) = 2(1+ \cos\theta)^2 \geq 0$

Because for all $n \geq 1$ we have $\Lambda(n)n^{-\sigma} \geq 0$, we have

Now let


 * $\eta(s) = \zeta(s)^3\cdot \zeta(s+it)^4\cdot \zeta(s+2it)$

Then the above computation has shown that


 * $\displaystyle\Re\left( \frac{\eta'(s)}{\eta(s)} \right) \leq 0$

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

Suppose that $1+it$ is a zero of $\zeta$ of order $d \geq 1$.

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


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

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


 * $\displaystyle \frac{\eta'(s)}{\eta(s)} \sim \frac{4d -3}{s-1}$

as $s \to 1^+$. Since $\displaystyle \Re\left( \frac{4d -3}{s-1}\right) \to + \infty$ as $s \to 1^+$, it follows that


 * $\displaystyle \Re\left(\frac{\eta'(s)}{\eta(s)}\right) \to \infty$

as $s \to 1^+$. But we have already shown that


 * $\displaystyle\Re\left( \frac{\eta'(s)}{\eta(s)} \right) \leq 0$

a contradiction.