Calculating the Zeroes of the Riemann-Zeta function

Question

Wikipedia states that

The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: The real part of any non-trivial zero of the Riemann zeta function is $\frac{1}{2}$.

What does it mean to say that $\zeta(s)$ has a $\text{trivial}$ zero and a $\text{non-trivial}$ zero. I know that $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$ what wikipedia claims it that $\zeta(-2) = \sum_{n=1}^{\infty} n^{2} = 0$ which looks absurd.

My question is can somebody show me how to calculate a zero for the $\zeta$ function.

Answer 1

You are going to need a bit of knowledge about complex analysis before you can really follow the answer, but if you start with a function defined as a series, it is frequently possible to extend that function to a much larger part of the complex plane.

For example, if you define $f(x)=1+x+x^2+x^3+...$ then $f$ can be extended to $\mathbb C\setminus \{1\}$ as $g(x)=\frac{1}{1-x}$. Clearly, it is "absurd" to say that $f(2)=-1$, but $g(2)=-1$ makes sense.

The Riemann zeta function is initially defined as a series, but it can be "analytically extended" to $\mathbb C\setminus \{1\}$. The details of this really require complex analysis.

Calculating the non-trivial zeroes of the Riemann zeta function is a whole entire field of mathematics.

Answer 2

Copied from Wikipedia:

For all $s\in\mathbb{C}\setminus\{1\}$ the integral relation $$\zeta(s) = \frac{2^{s-1}}{s-1}-2^s\!\int_0^{\infty}\!\!\!\frac{\sin(s\arctan t)}{(1+t^2)^\frac{s}{2}(\mathrm{e}^{\pi\,t}+1)}\,\mathrm{d}t,$$ holds true, which may be used for a numerical evaluation of the Zeta-function. http://mo.mathematik.uni-stuttgart.de/kurse/kurs5/seite19.html

Answer 3

Regarding your question on how to calculate zeros of $\zeta(s)$...

The algorithm below will converge to a nontrivial zero of $\zeta(s)$ along the line $s=1/2+t \in \mathbb{C}$.

$\mathbf{Newton-Raphson}$ $\mathbf{Algorithm}$ $\mathbf{for}$ $\zeta(s)$: Given an initial $t_{k} \in \mathbb{R}$, iterative solutions $t_{k+1}$ converge to nontrivial zeros of $\zeta(s)$,

$$t_{k+1}=t_{k}-\frac{2 i}{\frac{16 i t_{k}}{1+4t_{k}^2}+\log_{e}(\pi) - \psi(1/4+i t_{k}/2)-\frac{2 \zeta’(1/2+i t_k)}{\zeta(1/2+i t_k)}}.$$

Note that $\psi(s)$ is the di-gamma function and that $t_k$ is the imaginary part of the root $s=1/2+i t_k$, with $\zeta(1/2+it_k)\approx 0$.