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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.

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    As Ginger mentions, one uses the Riemann-Siegel formula *numerically* to compute the nontrivial zeroes (there are no known closed forms for them).2012-04-20

4 Answers 4

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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.

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    Analytic continuation is nevertheless$a$well-defined concept (see e.g. wiki), but certainly the adjective "artificial" wasn't carefully chosen either;)2012-04-22
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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

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Here's an extension method in c# to calculate the zeroes for Re(s) > 0. However, it is not very efficient for large values of t. Note, .5 in the calculation is the Zeta(1/2+it). Try any other number and you will not get a zero.

Also, one could easily modify the function to return an IEnumerable and the user could create a query/filter against each term in the infinite sum. I found it interesting to plot each term on a graph and watch it converge in the plane. The zeroes are where the graph comes back to the origin. The Complex type is found in the System.Numerics namespace.

    ///      /// Calculates the converged point for a Dirichlet series expansion.     ///      /// imaginary part of s. The first zero is at 14.134725     /// Use a higher number to find more accurate convergence.     ///      public static Complex CalcZetaZero(this double t, int numberOfTerms)     {         var range = Enumerable.Range(1, numberOfTerms);         var zetaZero = Complex.Zero;          foreach (int n in range)         {             var direction = n % 2 == 0 ? Math.PI : 0;             var newTerm = Complex.Exp(new Complex(-Math.Log(n) * .5, -Math.Log(n) * t + direction));             zetaZero += newTerm;         }          return zetaZero;     }