Bound $\ell_2$ norm of an embedding $\sigma_i \in\mathsf{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$

Question

It is well-known that $\mathsf{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) = \{\zeta_n^i \mid \gcd(i, n) = 1\}$. Let $\sigma_i : \zeta_n\mapsto \zeta_n^i$. I am curious about how to bound the real and imaginary parts of $\sum_{i\in(\mathbb{Z}/n\mathbb{Z})^\times} \zeta_n^{2i}$. Focusing on the real part for now, there is the obvious bound of $\zeta_n^{2i} \leq 1$, so the sum is at most $\varphi(n)$. Can this bound be improved much (for general $n$)? For example (in Landau notation) is this bound $o(\varphi(n))$, or $\Theta(\varphi(n))$?

Answer 1

The bound can actually be made exact. The relevant quantity is an instance of Ramunajan's sum

$$ c_q(n) = \sum_{a\in(\mathbb{Z}/q\mathbb{Z})^\times}\exp(2\pi i \frac{a}{q}n) = \mu\left(\frac{q}{\gcd(q,n)}\right)\frac{\varphi(q)}{\varphi\left(\frac{q}{\gcd(q,n)}\right)},$$ where $\mu(n)$ is the Mobius function (the second equality is with what is known as "von Sterneck's arithmetic function"). As my setting cares about $c_n(2)$, the relevant sum is $$ \mu\left(\frac{n}{\gcd(2,n)}\right)\frac{\varphi(n)}{\varphi\left(\frac{n}{\gcd(n,2)}\right)}. $$ From this expression, it is clear for which $n$ we can hope for a gap between $c_n(2)$ and $\varphi(n)$.