# 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

This needs considerable tedious hard slog to complete it.In particular: First it needs to be established that those points are in fact zeroes. This follows directly and trivially from Riemann Zeta Function at Non-Positive Integers, but needs to be made explicit here.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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.

$\blacksquare$

## Also see

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

## Sources

- 1983: François Le Lionnais and Jean Brette:
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