Currently, I'm attending an Analytic Number Theory course, and in the lecture notes I've come across the following statement:
Does anyone know how to prove this, or at least can give me a reference? Moreover, I'm also confused by the variable $a$, which appears in (5.4) and in (5.5) again. They should be different, right?
Would be very happy with any hint! :)
Best, python15
The Dirichlet characters $\bmod q$ are an orthogonal basis of $$\{ v\in \Bbb{C}^q, \gcd(a,q)\ne 1\implies v(a)=0\}$$ With $v(a) = e(a/q)1_{\gcd(a,q)=1}$ then $$\tau(\chi)=\langle v,\overline{\chi}\rangle, \qquad v=\sum_{\chi\bmod q}\overline{\chi}\frac{ \langle v,\overline{\chi}\rangle}{\|\overline{\chi}\|^2}$$