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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

We used the wonderful table of structure and generators of (Z/nZ)* in

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

  1. 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, <http://www.di-mgt.com.au/dirichlet-character-generator.html>, {date accessed}.

References

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This page last updated 17 June 2017.