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A Dirichlet character $\chi_1$ of modulus $N_1$ is induced by a Dirichlet character $\chi_2$ of modulus $N_2$ dividing $N_1$ if $\chi_1(n)=\chi_2(n)$ for all integers $n$ coprime to $N_1$.

If $N_1$ and $N_2$ have the same prime divisors then $\chi_1=\chi_2$, but in general $N_1$ may be divisible by primes $p$ not dividing $N_2$ in which case $\chi_1(p)=0$ but $\chi_2(p)\ne 0$ and the characters differ.

Every Dirichlet character $\chi$ is induced by a unique primitive Dirichlet character $\tilde\chi$ whose modulus is the conductor of $\chi$, and for every integer $N'$ divisible by this conductor there is a unique Dirichlet character $\chi'$ induced by $\tilde\chi$ which satisfies $\chi'(n)=\chi(n)$ for all integers $n$ coprime to both $N$ and $N'$.

Knowl status:
  • Review status: beta
  • Last edited by Andrew Sutherland on 2018-12-25 18:05:29
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