# A Dirichlet character table generator

This page has an on-line calculator which generates a table of Dirichlet characters for modulus $k\leq 62$.

## Dirichlet characters - some background

A Dirichlet character $\chi$ is an arithmetic function from the integers to the complex numbers. The concept was introduced by P.G.L. Dirichlet [DIRE] in his proof of Dirichlet's theorem that there are an infinite number of primes in arithmetical progressions with first term $h$ and common difference $k$ where $(h,k)=1$. [Note 1]

By definition, a Dirichlet character modulo $k$ is a completely multiplicative function, periodic with period $k$, which is not identically zero and which equals zero if its argument is not coprime with $k$ [NUME] [SOUND].

That is, $\chi(m)\chi(n) = \chi(mn)$ and $\chi(n) = \chi(n+k)$ for all integers $m, n$; $\chi \not\equiv 0$, and $\chi(n)=0$ if $(n,k)>0$.

From this definition, it follows that $\chi(1)=1$ and that each $\chi(n)$ is a $\phi(k)$th root of unity when $(n,k)=1$, where $\phi$ is Euler's totient function, the number of positive integers less than or equal to $k$ that are coprime to $k$. There are exactly $\phi(k)$ characters for a given modulus $k$,

The principal character, denoted $\chi_1$, has value one if $n$ and $k$ are coprime and value zero otherwise. Characters other than the principal character are known as nonprincipal characters and are denoted $\chi_2, \chi_3, \ldots, \chi_{\phi(k)}$. This latter numbering is arbitrary.

The nonprincipal characters may have complex values. There is, however, always at least one real-valued nonprincipal character.

We can arrange the nonzero values of $\chi$ in a square matrix $A$ of size $\phi(k)$ with the columns equal to the values of $n$ coprime to $k$. The rows are then the nonzero values of the characters $\chi_1, \chi_2, \ldots ,\chi_{\phi(k)}$. The elements of the first row and first column all equal one and so sum to $\phi(k)$, and we find that the sum of the elements of the other rows and columns is always zero.

## The generator

The generator is here. You can select values of $k$ between 2 and 62 and it will show the resulting table of non-zero values for Dirichlet characters modulo $k$.

It also gives the table in "text" form should you need that, and a version that can be cut and pasted into a Latex document. You might find that very useful for assignments.

The results in the generator verify against the tables in

- Numericana.com
- Table Of Dirichlet Series For Small Moduli, Richard J. Mathar.
- And, of course, Apostol, Introduction to Analytic Number Theory.

**Z**/n

**Z**)* in

- Wikipedia, Multiplicative group of integers modulo n

## More theory

How we did it using group theory: Generating Dirichlet Characters (PDF, 146 kB). Can you think of a reason we only went up to $k=62$?

## Notes

- We use the notation $(x,y)$ to denote the greatest common divisor of $x$ and $y$. This can also be denoted $\text{gcd}(x,y)$.

## How to cite this page or the generator

David A. Ireland, A Dirichlet character table generator, D.I. Management Services Pty Ltd, <https://www.di-mgt.com.au/dirichlet-character-generator.html>, {date accessed}.

## References

**[APOS]**Tom M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.**[DIRE]**P.G.L. Dirichlet, Vorlesungen über Zahlentheorie, Vieweg, 2nd ed., 1871.**[MATHAR]**Richard J. Mathar, Table Of Dirichlet Series For Small Moduli, 2010, <http://arxiv.org/abs/1008.2547>**[NUME]**Gérard P. Michon, Number Theory: Dirichlet characters, <http://www.numericana.com/answer/numbers.htm#characters>**[SOUND]**K. Soundararajan, Dirichlet's Theorem On Primes In Progressions, III. Available online at Dirichlet3.pdf**[WIKI]**Wikipedia: The Free Encyclopedia. Multiplicative group of integers modulo n, <http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n>

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*This page last updated 2 March 2021. *