Definition:Riemann Zeta Function/Zero

Definition

The zeroes of the Riemann zeta function are, according to the conventional definition of the zero of a (complex) function, the points $s \in \C$ such that:

$\map \zeta s = 0$

Trivial Zeroes

The trivial zeroes of the Riemann zeta function $\zeta$ are the strictly negative even integers :

$\set {n \in \Z: n = -2 \times k: k \in \N_{\ne 0} } = \set {-2, -4, -6, \ldots}$

Nontrivial Zeroes

The nontrivial zeroes of the Riemann zeta function $\zeta$ are the zeroes of $\zeta$ that are not trivial.

Sources

• Weisstein, Eric W. "Riemann Zeta Function Zeros." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html