Trivial Zeroes of the Riemann Zeta Function

Theorem

Suppose $\rho$ is a zero of the Riemann zeta function not contained in the critical strip:

$0 \le \Re \left({s}\right) \le 1$

Then:

$s \in \left\{{-2, -4, -6, \ldots}\right\}$

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

Proof

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

Therefore, the completed Riemann zeta function:

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

has the same zeros as $\zeta$.

Additionally by Functional Equation for Riemann Zeta Function, we have $\xi \left({s}\right) = \xi \left({1-s}\right)$ for all $s \in \C$.

Therefore if $\zeta \left({s}\right) \ne 0$ for all $s$ with $\Re \left({s}\right) > 1$ then also $\zeta \left({s}\right) \ne 0$ for all $s$ with $\Re \left({s}\right) < 0$.

Let us consider $\Re \left({s}\right) > 1$. We have:

$\displaystyle \zeta \left({s}\right) = \prod_p \frac 1 {1 - p^{-s}}$

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

Therefore, we have:

$\displaystyle \zeta \left({s}\right) \prod_p \left({1 - p^{-s}}\right) = 1$

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

$\displaystyle \prod_{n=2}^\infty \left({1 - n^{-s}}\right)$

which converges absolutely since the zeta sum $\displaystyle \sum_{k=1}^\infty k^{-s}$ converges absolutely.

Hence:

$\displaystyle \prod_p \left({1 - p^{-s}}\right)$

converges absolutely, and so by the fact that:

$\displaystyle \zeta \left({s}\right) \prod_p \left({1-p^{-s}}\right) = 1$

we know $\zeta \left({s}\right)$ can't possibly be zero for any point in the region in question.

$\blacksquare$

Notes

The non-trivial zeroes are those zeroes that occur in the so-called "critical strip" $\left\{{s : 0 \le \Re \left({s}\right) \le 1}\right\}$ and are the subject of much study. It is known that there are no zeroes on the boundary of the critical strip, ie, $\Re \left({s}\right) = 1 \lor 0$. The Riemann Hypothesis states that all zeroes in this region fall on the line $\Re \left({s}\right) = \tfrac 1 2$, called the "critical line." It has been shown that an infinite number of zeroes lie on the line, and the proportion of zeroes not falling on the line is zero. For more information on the importance of the Riemann hypothesis, see the notes of the Prime Number Theorem.