Suppose I have a real multiplicative character module $q$, say $\chi$. Could some one please explain me why the following is true: given $n$, $$ \prod_{p^{\alpha} || \ n} (1 + \chi(p) + ... + \chi(p^{\alpha}) \geq 0 ? $$ Thank you!
2026-04-11 13:05:47.1775912747
A basic question on real multiplicative character
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Your character, being real-valued, takes values in $\pm 1$. It suffices to prove it for $n=p^a$. If $\chi(p)=1$ then the sum is clearly positive. If $\chi(p)=-1$ then $1+\chi(p) + \dots+ \chi(p^a) = 1-1 + \dots + (-1)^a$ is either $0$ or $1$ according to the parity of $a$. If $\chi(p)=0$ then the sum is just $1$.