Verify Complex Solutions of Riemann Zeta Function

complex-analysisriemann-zeta

I've recently been trying to visualize the Riemann Zeta complex function by graphing it on a 3D plot.

I wrote some code to compute the value of the following integral representation of the Riemann Zeta function, which should be defined in Re(Input) $\neq$ 1:

$$\zeta(s) = \frac{2^{s-1}}{s-1} -2^{s} \int_{0}^{\infty}\frac{\sin(s\arctan(x))}{(1+x^{2})^{\frac{s}{2}}(e^{\pi x}+1)}dx$$

Where s is the complex Input.

Through the use of GNUPlot's code libraries, I graphed the following:

  • Re(Input) – X-axis
  • Im(Input) – Y-axis
  • Re(Output) – Z-axis
  • Im(Output) – Color

As far as I understand, this is one of the standard ways to visualize & graph complex functions.

With this setup I'm able to generate some plots:

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Plots above depicts the function on -5 < Re(Input) < 5 and -5 < Im(Input) < 5. (forgive the messed up colorbar label on the right, it appears that C++'s GNUPlot library is quite old…)

enter image description here
enter image description here

Plots above depicts the function on -30 < Re(Input) < 30 and -30 < Im(Input) < 30.

So far, so good. However, I want to be able to verify whether my code produces the correct answers for this representation of the RZ function.

I'm able to compare some of my answers to some of the known answers out there, such as any answer with Re(Input) > 1, real negative integers and discovered nontrivial zeroes, and those all appear to check out. However, apart from those, I haven't really been able to find a good way of reliably verifying the rest of my answers here. Most popular online function graphing calculators either fail to compute that integral function (such as Desmos), are unable to accept complex inputs (such as GeoGebra), or only have static plots of it (such as Wolfram).

Since I'm not a mathematician, and since I'm not aware of all the possible resources out there or methods for verifying solutions for complex functions, I wanted to ask that here.

What would be some good ways of verifying solutions to complex functions, such as the Riemann Zeta function? Is there a tool that is used when computing complex functions such as these, for verifying solutions?

Thanks for reading my post, any guidance is appreciated.

Best Answer

You can do it in Wolfram Alpha (or Mathematica if you have access to that). Like this, for example:

Plot3D[Re[Zeta[x+I y]],{x,-5,5},{y,-5,5}]

If you're a Python programmer, you can use the mpmath library to compute the zeta function for complex arguments.

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