Let $\chi$ be a Dirichlet character modulo $q$. Let $e(x)=\text{exp}(2\pi i x)$ and $\tau(\chi)$ be the Gauss sum $\tau(\chi)=\sum_{m \,\text{mod}\, q}\chi(m)e(m/q)$. For any $n$ with $\gcd(n,q)=1$, we know that $$ \chi(n) = \frac{1}{\tau(\overline{\chi})} \sum_{m \,\text{mod}\, q} \overline{\chi}(m)\,e\!\left(\frac{nm}{q}\right). $$ If $\chi$ is primitive, then the equation holds for all $n\in\mathbb{Z}$. If $\chi$ is not primitive and $\gcd(n,q) > 1$, then $\chi(n)$ is zero while the right hand side may not be. Is there a formula similar to this one, that writes a multiplicative character as a sum of additive characters, that holds for all multiplicative characters, not just the primitive ones?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.