Trivial Zeroes of Riemann Zeta Function are Even Negative Integers

Theorem
Let $\rho = \sigma + i t$ be a zero of the Riemann zeta function not contained in the critical strip:
 * $0 \le \map \Re s \le 1$

Then:
 * $s \in \set {-2, -4, -6, \ldots}$

These are called the trivial zeros of $\zeta$.

Proof
First we note that by Zeroes of Gamma Function, $\Gamma$ has no zeroes on $\C$.

Therefore, the completed Riemann zeta function:


 * $\map \xi s = \dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \, \map \zeta s$

has the same zeroes as $\zeta$.

Additionally by Functional Equation for Riemann Zeta Function, we have:
 * $\map \xi s = \map \xi {1 - s}$

for all $s \in \C$.

Therefore if:
 * $\map \zeta s \ne 0$

for all $s$ with $\map \Re s > 1$, then also:
 * $\map \zeta s \ne 0$

for all $s$ with $\map \Re s < 0$.

Let us consider $\map \Re s > 1$.

We have:


 * $\ds \map \zeta s = \prod_p \frac 1 {1 - p^{-s} }$

where here and in the following $p$ ranges over the primes.

Therefore, we have:
 * $\ds \map \zeta s \prod_p \paren {1 - p^{-s} } = 1$

All of the factors of this infinite product can be found in the product:


 * $\ds \prod_{n \mathop = 2}^\infty \paren {1 - n^{-s} }$

which converges absolutely.

This follows because from Convergence of P-Series:
 * $\ds \sum_{k \mathop = 1}^\infty k^{-s}$ converges absolutely.

Hence:


 * $\ds \prod_p \paren {1 - p^{-s} }$ converges absolutely.

So by the fact that:


 * $\ds \map \zeta s \prod_p \paren {1 - p^{-s} } = 1$

we know $\map \zeta s$ cannot possibly be zero for any point in the region in question.

Also see

 * Riemann Hypothesis: all zeroes in the critical strip fall on the critical line $\map \Re s = \dfrac 1 2$.


 * Critical Line Theorem