Back to Riemann, Part 18


So, what does the gamma function look like in Wolfram Alpha?

From an Excel animation viewpoint, the really interesting part of the graph is when you get to about x=6. From the above plots, you can see that there are a couple small poles in the x< region, then the big one at x=0, which each provide a little activity leading up to x=6, but otherwise, the map is flat for the left hand half of the complex plane.

The animation also depends on which variable you treat as the line segment. In the first pass, the segment for x=-10 to 10 was calculated for varying values of y (for y=-10 to 10). In the second pass, it was the other way around. The second animation is much more interesting. I also zoomed in on the segment for the area from -0.5 to 1. Note that for the animation where the y segment is moved around, the segment disappears due to clipping as the results from gamma() get larger. The unraveling linked spirals are kind of hypnotic, but there aren’t quite enough data points between x=6 and x=8 to generate enough frames for looping them in movie maker.

Direct youtube link

And this is pretty much where I’m going to call it quits with gamma and zeta.

A final few words on the book itself. The Riemann Hypothesis, by der Veen and de Craats is a short book, at only 144 pages, but it is highly readable, and much more approachable than any of the sources I looked at. They include a lot of exercises that can help you link the different concepts together, as well as showing how to use Wolfram Alpha for doing the computer grunt work. I don’t know why it’s suddenly so expensive on Amazon, but if you can find a copy used for $15, it’s worth the price. Recommended if you want a decent introduction to Riemann and the zeta function. I’ve gotten about 3 month’s worth of activities from it, which isn’t bad for a little book like this.

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