# Plotting Zero-free regions

So I am working on a project for proving different types of zero-free regions, I needed to plot some zero free regions using sage and cannot think any obvious ways to do it. I am still pretty new to sage and computational math in general but I needed some guidance as to what I should do. I am trying to plot the zero free region by de la Vall ́ee Poussin which is $1-\beta \leq \frac{c}{\log(t)}$

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what gives region_plot? ?

More generally, the reference manual may come in handy...

( 2022-03-29 08:19:46 +0200 )edit

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Maybe something like

var('sigma,t')
c=0.3
region_plot((1-sigma)-c/log(2+abs(t))<= 0, (sigma, 0, 1), (t, -100, 100),aspect_ratio=1/99)


will do the trick. Notice that I have changed the argument of the log a bit to make it non-negative. The form of the Hadamard-dVP region you have stated is only nontrivial for $\lvert t \rvert$ sufficiently large anyway.

If you want the symmetric version you may try

region_plot((1-sigma)-c/log(2+abs(t))<= 0, (sigma, 0, 1), (t, -100, 100),aspect_ratio=1/99)+region_plot((sigma)-c/log(2+abs(t))<= 0, (sigma, 0, 1), (t, -100, 100),aspect_ratio=1/99)

more

I was able to do it by calculating points and the plotting using scatter_plot but I like this way better but I wanted to ask how do I scale the x axis on this plot because if I go for 100 on the y axis then x axis appears as 0. Because using scatter plot I am getting a very nice visualization of the zero-free region, but this is much cleaner code wise. Thank you for your answer

( 2022-03-30 03:37:05 +0200 )edit

You can play around with aspect_ratio. I have updated the answer accordingly. If it solves the original question, you can mark it as accepted by clicking the "accept" button next to it (the button with a check mark). This will mark the question as solved in the list of questions.

( 2022-03-31 07:59:46 +0200 )edit

Thank you very much @mrisager

( 2022-03-31 22:10:34 +0200 )edit