Dirichlet Characters

In mathematics, a character is a well-behaved function from a group to a field. In number theory, the most common characters are Dirichlet characters, which are functions from the integers to the unit circle.

Image of chi(5,4)
$\chi(5,4)$. DLD Sep 2020.

More precisely, a Dirichlet character \(\chi\) is a multiplicative function $$ \chi : (\mathbb{Z}/N\mathbb{Z})^\times \longrightarrow \mathbb{C}. $$

Every Dirichlet character has a label consisting of two integers, \(N\) and \(a\). The character \(\chi(N, a)\) (sometimes written \(\chi_N(a, \cdot)\)) is defined modulo \(N\) and has index \(a\).

The idea of plotting a Dirichlet character is a bit odd, since the "image" consists of several points on the unit circle. These plots would highly resemble various approximations of the circle --- mostly uninteresting. But a character can be interpolated to a function \(\chi : \mathbb{R} \longrightarrow \mathbb{C}\) through a Gauss sum $$ \tau(\chi) := \sum_{n \bmod N} \chi(n) \exp(2 \pi i n / N). $$ One can prove the identity $$ \chi(x) = \frac{\chi(-1) \tau(\overline{\chi})}{N} \sum_{n \bmod N} \overline{\chi}(n) \exp(2 \pi i n x / N), $$ and note that this function makes sense for all \(x \in \mathbb{R}\). On integers, it agrees with the typical Dirichlet character.

Now it is possible to plot many characters. These are the characters mod \(22\), excluding the trivial character.

Nontrivial Dirichlet characters mod 22
$\chi(22,\cdot)$. DLD Sep 2020.

Some characters look very simple.

Image of chi(21,4)
$\chi(21,4)$. DLD Sep 2020.
Image of chi(21,10)
$\chi(21,10)$. DLD Sep 2020.

Others look more complicated.

Image of chi(200,119)
$\chi(200,119)$. DLD Sep 2020.

Others look downright busy.

Image of chi(201,179)
$\chi(5,4)$. DLD Sep 2020.

But as one continues to make and study these images, patterns emerge. A curtain hides an underlying image --- subsequent characters shed light, but the image remains obscured.

A sequence of characters
Emergence. DLD Sep 2020.

Or there may be a feeling like someone is watching you.

A sequence of faces
Faces. DLD Sep 2020.

Even the busy character above seems to be one part in a sequence. The next term might look like the following.

Image of chi(256,51)

For characters with larger moduli, the images take on a different facet of beauty.

Image of chi(1001, 9)
$\chi(1001,9)$. DLD Sep 2020.

The clustering around the center and around an apparent circle are not coincidental. At integer inputs $n$, $\chi(n)$ will either be $0$ or lie on the unit circle. In many images, the tension comes from the the apparent randomness and the need to return to the circle.

Prints of high-res versions of these images are available. Contact me at mixedmathart@lowryduda.com for more.

Image of chi(202,201)
$\chi(202,201)$. DLD Sep 2020.
Image of chi(222,187)
$\chi(222,187)$. DLD Sep 2020.
Image of chi(248,247)
$\chi(248,247)$. DLD Sep 2020.
Image of chi(254,253)
$\chi(254,253)$. DLD Sep 2020.
Image of chi(290,161)
$\chi(290,161)$. DLD Sep 2020.