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Let $$\chi_q(n,\cdot)=\prod_{p|q}\chi_{p^e}(n,\cdot)$$ be the unique factorization of the Dirichlet character $\chi_q(n,\cdot)$ into characters of prime power modulus $p^e$ under the Conrey labeling system. The parity of $\chi_q(n,\cdot)$ is the sum of the parities of the Dirichlet characters $\chi_{p^e}(n,\cdot)$, which can be computed as follows:

  • for $p>2$, the character $\chi_{p^e}(n,\cdot)$ is even if and only if $n$ is a square modulo $p$;
  • for $p=2$ the character $\chi_{p^e}(n,\cdot)$ is even if and only if $n$ is a square modulo $4$.
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  • Review status: reviewed
  • Last edited by Alina Bucur on 2018-07-04 21:43:35
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